PRIRCE
GIFT or
lirs. G. N, LeTTis
t
SHORT TABLE OF INTEGRALS
BT
B. O. PEIRCE
HoLLis Professor of Mathematics and Natural Philosophy in
Harvard University
SECOND REVISED EDITION
GINN AND COMPANY
BOSTON • NEW YORK • CHICAGO • LONDON
ATLANTA • DALLAS • COLUMBUS • SAN FRANCISCO
COPYRICIIT, ISVd, 1910
By GINN and COMPANY
ALL EIGHTS RESERVED
522.2
1
Since I cannot hope that these formulas are wholly free
from misprints, I shall be grateful to any person who will
call my attention to such errors as he may discover.
B. O. PEIRCE,
Harvard University, Cambridge.
GIFT
tH)t Stfjenatum jPrees
I'.IW" AMI CllMl'ANY • Vlli).
PKIETURS • liUSTuN • U.S.A.
Q A- =s / o
HA If '
TABLE OF INTEGRALS.
-o-o>0<o°-
I. FUNDAMENTAL FORMS.
1. I adx = ax.
2. \ ctf(x) dx = a ( f(x) dx.
3. f— = logx. noga: = log(-x) + (2A:+l)7r/.]
J x
oc'^dx = ' when vi is different from — h
m -\- 1
5. Ce'^dx = e'.
6. C a^ log adx = a",
7 I _^^- — = tSLU-^x, or — ctn-^a;.
. = sin~^cc, or — cos ^x
dx
/ dx
X Va;^ — 1
10. . ,
V2 X — X
= sec-^cc, or — esc ^x.
-1
dx
— = versin-^a;, or — coversin-^a;.
-J'2x-x^
M623267
FUNDAMENTAL FORMS.
1. I COS xdx = sin x, or — coversin x.
2. I ^va.xdx = — cos x, or versin x.
3. I ctn ic c?x = log sin x.
4. I tan xdx = — log cos x.
5. I tan X sec a; (fx = sec 05.
6. I sec^icc?ic = tancc.
7. I csc^ajt^x = — ctnx.
In the following formulas, u, v, w, and y represent any
unctions of x :
8. i (u-\-v + w + etc.) dx = i udx + i vdx + i wdx-\- etc.
9 a. i udv ^= uv — i V du.
_ /• c?y , f du ^ - '
9 6. \ u—-dx = uv — \ v-r- dx.
J dx J dx
.o.//(.)^=//Mf^
dx
RATIONAL ALGEBRAIC FUNCTIONS.
II. RATIONAL ALGEBRAIC FUNCTIONS.
A. — Expressions Involving (a + hx).
The substitution oi y ov z for x, where y = a -\- bx,
z = (a + bx) / X, gives
21. ("(a + bx)^dx = ijy^dy.
22
23
. Cx{a + bxYdx = y,yf {y - «) dy-
r x'^dx ^ _i_ r {y-(^Tdy .
^ J Ca + Z^x)"* i»+V
25
(a + Z^x)"* i"+V 2/'"
■J a;«(a4-te)'" «""+"- V
Whence
dx 1
(Zx _
+ bxf "
dx 1
27. fy-
J (a
r dx ^ 1
^^- J (a + ix)« 2 6(a + te)=
6 RATIONAL ALGEBRAIC FUNCTIONS.
r ^^ix _ 1 r 1 a ~|
J {<i + te)' ^z-' L « + ^* 2 (a + bxy\
32. /;f^ = ^3 \i (* + ^^)' - 2a (a + hx) + a" log (a + bx)\
34,
35
/ dx _ 1 ■■ a + ^>x _ *
a; (a + 6j:;) a x
/dx _ 1 _ \_ , rz-t-^A-r
xfa+^a;)^ a(a+bx) a^
(a+bxf a{a+bx) a^ x
37. C (a+bxY {a'+b'x)'"dx = - — ■ — -y ( (a+bxy + ' (a'+b'xf
J ^ ^ ^ ' {in-\-n-\-\) b \
— m(ab'-a'b) C (a-^-bx)" (a' + b'xy-hlxj
r {a + bxydx 1 f (a + hTY + ^
' J (a'
I _L /,' ,• \in — L
{a' + b'x)'" (m — l){ab' - a'b) \{a' + b'x)
r(a + bxYdx
+ (m — n — Z)b I -r-; z-^: r
^ 1 / (a-\-bxY '
~ (m — n-l)b'\(a' + b'xy"-^
+ n
^ ^J {a' + b'x)'" J
_ i f (a + bx)" r (a + bx)''^^dx \
* /'—^^^— =--'- + - log ^^i-^ .
J x\a + bx) ax a"^ " x
39
RATIONAL ALGEBRAIC FUNCTIONS
dx 1 , ft' + Vx
(a + bx) {a' + h'x) ~~ ab' -a'b' °^ a-\- bx
dx
40./
(a + bxy {a' + b'x)""
= I ( 1
(m - 1) (ab' - a'b) \{a + bx)"-^ (a' + b'x)"'-'
dx ^
41
- (m + n-2) bC
f
(a + bxy (a' + b'xy"-\
xdx
(a + bx) (a' + b'x)
42, j
43
(a + bx)\a' + b'x)
= _J— f^- + — ^— log ^^^±^\
ab' — a'b \a + bx ab' — a'b a + bx J
r xdx
(CI + bxf (a' + b'x)
— a a' . a' + b'x
TTTz log ■
b (ab' - a'b) (a + bx) (ab' - a'by ^ a + bx
x'^dx a^
(a + bxy (a' + b'x) ~~ b' (ab' - a'b) (a + bx)
1 r«'% / r . 7-r N , a(ab'-2a'b) ^ . , . .1
— ^ [y log (a' + b'x) + -^—j, '- log (a + bx)j
/I n 2+i
"^ («&' -
.^ C dx n , ,-. ^~
(a + OXJn ^ ^
8 RATIONAL ALGEBRAIC FUNCTIONS.
B. — Expressions Involving (a + 5x").
r dx 1, _iX 1 . _, X
47. I -T", — -„ = -tan ^-=-sin i— ===•
.^ C dx It c + x C dx 1 . .-r — c *
48. I -^ 5 = — log > \ —^ i = 77-log — — -•
J c^ — x^ 2 c c — xjx^ — cr 2 c x-\-g
50. I — r^-i = — 7=log--= 7=j if a>0, b<0.
J a + bx^ 2V^^ V^-a;V-6
/* <?a; _ a; 1 /* c?x
J (a + bxy ~ 2a(a + bx') 2aJ a + bx^'
r dx _ 1 ^ _i_ ^ ^^ ~ 1 r rfa
■ J (a + Z/x2)"*+i - 2 7/ia (a + bx^)"" 2 ma J {a + bx^)"^
" ./ a; ((X + 6x^) 2 a a + bx'^
/x^dx _x a r dx
a + bx^ b bJ a + bx^
56
r dx _ _ j_ _ ^ r_^_
'J cc^ (a + ix'^) ax a J a + 6x^
, a;^c?aj — x . 1 /* r/.r
58.
r x^dx _ — X '^ C
(a -I- bxy'' + ^ 2mb(a + bx^"" 2 mbJ (a + bx^)'»
r dx -^ c ^^ ^ r— -^^— ■
J x\a + ^<x2)'» + i ~ a J x'^{a + ix^)*" a J (a + 6x')'"+*"
• p dx 1
y^-s-"--©^ /i^.-i-."-©-
RATIONAL ALGEBRAIC FUNCTIONS.
r«
-ii
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rSli
rS,
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II
11
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tH
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II
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^
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10 RATIONAL ALGEBRAIC FUNCTIONS.
C. — Expressions Involving (a -{- bx + cx^).
Let X = a + bx -\- cx^ and y = 4 ac — b"^, then
„„ rdx 2 ^ .2cx + b 2 ^ , .2cx-\-b
67. I ^ = —7= tan-^ ■ y^- , or — —= ■ tanli-^ —
J A -\/q -yjq V— q V — q
„_ Cdx 1 - 2cx + b — V— a
68. I — = — =log 7^' when 7 <0.
V— 9' 2cx + b + V— 2-
rdx 2cx + b 2e rdx
6 c" /^(Za;
Z'
r dx _ 2cx-{-b 2{2n-l)c C dx
-^ Cxdx 1 - ^^ & /^(/a;
72 I = — lof X I
' • J X 2c ° 2cJ X
rxdx _ bx-\-2a b Cdx ^
J "X^ ~ Jx ^J x'
r xdx _ 2 a + bx b (2 n — V) r dx
■J A'" + i~~ nqX'' nq J X"'
„_ rx^ - a; i T ^ b^ — 2ac rdx
^«- J 5=5''-" = ^ ^ix + tJ x'
r a;'" dx x"'-^ n — ??^ + 1 ^> /* a:"'-^rfa;
Jx" + i~ (2 71 - 7M + 1) c.Y" 2?i-m + l"cJ X'" + i
m — 1 a rx"'~^dx
"*" 2 M - w + 1 ' c J X'' + i '
*dx
X
RATIONAL ALGEBRAIC FUNCTIONS. 11
^„ rdx 1 , x^ b rt
r dx __b_. ^ _ JL , /^i!_ _ i\ C—
J x^X'^2 a" ^^ x" ax'^\2a'' aJJx'
an C ^^ — 1 n + m — 1 h f* dx
J aj'^X^+i ~ ~ (?» — 1) ax"'-'X" " m — 1 a J a;"'-iA'«-
2n + m — \ c r dx
m — 1 aJ x"'-2X» + i
r_^ ^^ ^ C^^ 1 r dx
J ^^X'~2a(r^-l).Y«-l 2 a J X» "*" a J xX«-i'
/" (a' + ^>'cc)(Zx _ j; 2a'c-bb' rdx
^^•J X» ~ 2(w-l)cX"-''^ 2c J X»'
86. r (a' + i'a:)'" X« dx = — ^ — ^— ( ^-'C^^' + b'x)"'-'X"+
J ^ ' {in -\r 2 n At V) c\ ^ '
+ (m + n){2 a'c - 6^>') f(a' + 6'a;)'"-iX"c?a;
- (m - l)(a6'=' - aW + ca'^) ("(a' + b'xy'-^X«dx\
12 RATIONAL ALGEBRAIC FUNCTIONS.
r (a' + b'x)'"dx _ 1 f (b + 2 ex) (a' + b'xf
J J:» ~q{n-l)\ .Y"-i
-2(m-2n + 3)cJ ^ — -j;^^
+ m(2 a'c - bb')J ^ j;;^! J
_ 1 f b' (a' 4- ^''x)"-^
~ {m-2n + l)c\ X"-"^
+ (m-n) (2 a'c - i&') J ^^ ji
- (m - 1) (ab'' - aW + ca'') f ^^-^^^^^^~^\
J (ct' + i'x)"'
1 / - VX^
b"{m-l)\(a' + b'x)'—^
X'^-'^dx
+ n{bb'-2a'c)^—
(a' + b'x)"'-'^
. „ C X—'dx \
1 / 4- ^-'X"
RATIONAL ALGEBRAIC FUNCTIONS. 13
89 C—^^—
J (a' + ^»'x)"'X»
b'
n v(«'
(m -1) (ab'^ - aW + ca''') \{a' + ft'a;)"— ^X"-'
1 / 6'
2 {ab'^ - aW + ca'^) \{n - 1) (a' + ^»'a;)"'-U'"-»
+ (2 a'c - bV) (—TT-^. TF-
(m + 2n-^)b'^ r dx \
If aJ'2 - aW + ca'^ = 0,
dx
J Ca'
(a' + b'xyX"
1 (-
(bb' -2a'c)\(a'
(m + 71-1) (bb' - 2 a'c) \(a' + b'xyX''-^
+ (m + 2n-2)c f—r-nr^ — pf-Y
'^ ^ J (a' + b'x)"'-^X"J
D. — Eational Fractions.
Every proper fraction can be represented by the general
form :
f(x) ^ g,x"-' + gr,x"-^ + g.x'^-' + . . ■ + ^^
F(x) x" + kix"-^ + k^x"-- + • • • + A;„
If a, b, c, etc., are the roots of the equation F(x) = 0, so
that
F(x) = (x- a)P {x — by (x - cy • • -,
14 RATIONAL ALGEBKAIC FUNCTIONS.
then
F{x) {x-ay {x-a)P-^ (x-ay-^^ x-a
(x- by {x- by-' '^ (x- by--' '^ x -b
1 ^1 ^_ ^2 , ^3 |_ . . . 1 ^'r
(X — Cy (X — 6')*' ^ (X — C)''~^ iC — C
I ••• ••• ••• ••• ■••
where the numerators of the separate fractions may be deter-
mined by the equations
•)
_, , , f(x) (x -a)' , , ^ fix) (x -by ,
If a, J, c, etc., are single roots, then j? = gr =?•=••• = 1,
and
i''(a;) cc — a x — b x — c
The simpler fractions, into which the original fraction is
thus divided, may be integrated by means of the formulas :
. hdx _ r h d (inx -{- ti) _ h
r hdx _ r
' J (mx + 71)' J
(mx + ny J m(mx + n)' m(l — l) (mx + ii)'-'
and I ■ — = — log (mx -f n).
J mx + n ni '
RATIONAL ALGEBRAIC FUNCTIONS. 15
If any of the roots of the equation f{pc) = are imaginary,
the parts of the integral which arise from conjugate roots
can be combined, and the integral brought into a real form.
The following formula, in which i = V — 1, is often useful in
combining logarithms of conjugate complex quantities :
log {x ± yi) = 1 log {x" + if) ± i tan-^ ^-
The identities given below are sometimes convenient :
1 _ 1 r h _ b' "I
{a + bx'') {a' + b'x^) ~ a'b - ab' ' \_a + bx'' a' + b'x'^J
7)1 + nx 1
{k + Ix) {a + bx + cx^) al^ + chP' - bkl
Vl(ml — nk) c(nk — '>nl)x + (aln + ckm — blfn)~\
\_ k -{- Ix a -{- bx -{- cx"^ J
I + mx^
{a + te") (a' + i'^") a'b
1 r^^ — am a'm — b'I~\
— ab'' \_a + bx" a' + b'x" J "
1 ^ +JL^ + J^,
(x + a) {x + b) (x + c) X + a x + b x -\- c
where
B = - —Z TJ C =
{a — b){a — c) {b — c){b- a) (c - a) (c — b)
(x + a)(x + b)(x + c)(x + ;/) x + a x + b x + c x + g'
where
^1 = T, : TT T ' ^ = Z 7T~/ 7T7 7^ ' ^tc.
{b — a){c-a){g — a) {a - b){c — b){g — b)
16 IRRATIONAL ALGEBRAIC FUNCTIONS.
III. IRRATIONAL ALGEBRAIC FUNCTIONS.
A. — Expressions Involving Va + hx.
The substitution of a new variable of integration,
y = Va + bx, gives
2
2
^a^bxdx = — V(a + bxf
«« C I T- , 2 (2 a - 3 ^>a;) V(a + bxf
92. J xVa + bxdx = ^ ^g^^a ^"
«„ r , / — T^- , 2 (8 a^ - 12 abx + 15 b^x^) V(a + Z»a;)«
93. J xWa + Z-xc?x = -^ ^^^, '- ^
94. I dx = 2Va + Z^x + a I — ■
98./-
(/a; _2^a + bx
i + bx ^
96. I -7= = ^^r7-„ Va + bx.
Va + bx
/xdx 2(2 a — bx) I
■ I = ^ Q7.2 Va +
v./ 4- /ax 3 ^>2
„„ f a;2t/a; 2 (8 a^ _ 4 a^,a; + 3 ^»V) / —
97. I , = ^ .^ ,„ ^ Va + ^aj.
00 r ^^ 1 , / VaTTx — Va . „ ^ „
98. I — ■ = — ^ log ( , — ], for a >
99
a; Va + bx Va \ Va + ^a; + Va/
/c?a; 2 ^ , (a+^ —2 , , ia+bx
— ■ = . — tan-i \^ , or —-= • tanh-^ \
IRKATIONAL ALGEBRAIC FUNCTIONS. 17
dx Va + hx ^ C ^^
, «« r dx ■\Ja-\-hx b r d
100. , = 77- I 7=
•^ x^s/a + bx «^ ^^'^ xy/a
+ bx
2±n
4± n 2 ±n
102. J .(. + fa)*.& = p [_L__i L__LJ.
r a;^c?a; _ 2a;"'Vtt + to 2 ma r x'^-^dx
' -^ Va + bx (2 w + 1) i (2 ?/i + 1) bJ Va + ^
/* (/a; _ Va + ^>x (2?^-3)& T dx
' '^ x"V^i + bx~ (n-l)ax"-^ {2n-2)aJ x"-Wi
105. Jfc±M*: = , J(, + fa)"-i-=rf, + a/(^
n dx 1 /* (/a: b C dx
106.
n — 2
da;.
, /^ dx _ 1 C ___dx b^C
x{a + bxy^ x(a + ia;) 2 (a + te)2
107. ^ fix, V^^x)dx = '^fff^lL:-^, :J\z—-'dz,
where z" = a -\- bx.
m + n
/, , , !^ , n(a-}-bx) «
(a + bx) n dx = - \,,\
/m p
f(x, (a + bx)", (a + bx^, • • ')dx
where f = a + bx, and s is the least common multiple of n,
q, etc.
18
IRRATIONAL ALGEBRAIC FUNCTIONS.
B. — Expressions Involving Both Va + hx and Va' + b'x.
Let u — a + bx, v = a' + b'x, and k = ab' — a'b, then
dx
k + 2bv r- ^'
dx = — . .,, Vmv
110. r^^
/ -^ vdx _ 1 / — h_ C
V^ & 2bJ ^uv
xdx V?<v ab' + a'b C dx
/
111.
112. /
113./
4 bV ' "" 8 66'c; v^
^c dx 1 / — h r dx
\ uv
dx
bb' 2 iZ-
2
?^/:
'wy
log(V^»^;'w + ^»V?;)
2 ^ . I- b'u 2 ^ ,
= — , tan~ '■ \\—, J or — == tanh"
1 . , 2 bb'x + a'b + ab'
I sm- ^
114. f-
1 , b'V^-^/kb'
lop-
116,
dx
V^ ~ Vkb' ^^^ 6' vw + Vkb" " ^^kb
dx 2 V«
2 , ^;'V'w
tan~^ — ^
-^-kb'
/dx
v\uv
kV^
116. /.."vr,.. = (2;;^,(2 .— v;; + ;./^).
rV^^ 1 / 2Vw r dx \
(7/1 - 1) b' \ i;'"- 1 ^ ^ "J „m-l ^y
v"'-'^dx^
IRRATIONAL ALGEBRAIC FUNCTIONS. 19
/dx 1 / Vm , , ,. , r d-^ \
120. fv^u^-idx = — —. — — r-Y 2 y'^ + iM'-i
J (2 7H -j- 2n + 1)0 \
+ (2 w - 1) k Cv^'u^-^dxy
1^
(2n-l)k
121. Cv'"u-<'' + i^dx = — — ^ ,, , ( 2 y'« + i?t-("-J)
(2 m - 2 w + 3) b' Cv"'u-^"-^^dx j
— v"'u-^"-^^
(2n - 1)^''
+ mb' Cv'"-'^u-'^"-i''dx \
122. Cv-'^u^'^-i^dx = tA T-rA 2 ti"-^-^'"-'^
J (2 m — 2 ?i — 1)6' \
+ {2n- l)kCu''-^^v-"'dxj
— u^ — ip—^™ — ^^
(m- 1)^'''
+ (*^ - i)^ r«"-it'-('"-')rfa; j.
123. ry-'"%-("+»<^a; = — -— -( 2 7-('»-i>w-("-«
J (2 w - 1) ^ V
+ (2 m + 2 ?i - 3)b'Cv-"'ir <"-«c?a; j.
20 IRRATIONAL ALGEBRAIC FUNCTIONS.
C. — Expressions Involving ^x^^a? ani> ^a? — x^.
24. f Vx2 ±a?dx = \ [x Va;2 ^ci'^a? log {x + Vx^ ± a^)].*
X
dx . , cc .a;
— =r sip-t-, or— cos~^-'
28.
29
/c?a; 1 .a 1 , x
; = - cos~^ -1 or - sec ^-■
x^x^ — a? a X a a
r dx __i, A, + v.-±.^ y
-^ xVa-±a;2 « \ a; /
on C ^«^ =*= ^^ 7 /~^ ; 1 «. + Vft^ ± X^
i*"- I rfx = Va"* dr x'' — a lo£? ■
^ X - X
r V.r- - a^ /— : a
I — — dx = Vx^ — a^ — a cos -•
»^ X X
32
^ V
„„ /* xt^x /-^
33. 1 , = Vx2 - a".
*^ Vx^ — o?
34
fxVx^ia^^^x = iV(x2±a2)S.
35. rxVa--x2(^x = - ^ V(a2 _ a:2)3.
IRRATIONAL ALGEBRAIC FUNCTIONS.
21
136. C>/{x^±aydx
2 z
dx
dx
:X
= ^\ x-\/(x^±d
137. f-\{a^-x^
138. f
139. /-
140. /-
141 I ~
* J -».//^2 _ o,.2\3 -v/«2 _ ^a
»')]■
V(a;2 zh ay a^ Vx^i^
dJx X
iCf^a; — 1
xc?a; 1
142. fa; V(^2^^(ix = ^y/(x'±ay.
143. Cx^J^F^^^^dx = - |V(^2_ 3.2)6;
144. faj^ Vx=^ ± a^cZx
= ^V(x2±a^«rF|x-N/^^±^^-|log(a;+Vac*-ba?)
8
a;'e?x
145. Cx^y/^^
log 2 = sinh - J (^^) = cosh - 1 (^^7^) ; tanh- 1 2 = - i • tan- \ziy
* (See Note ou pages 20-21.)
22 IRRATIONAL ALGEBRAIC FUNCTIONS.
/Va^~±x^ dx ->Ja^ ± x^ IT dx
X Lx -^ a-Va^'zhx^
147, fx-Va^ + x^ (^x = (± 1 x2 - ^2^ a2) V(«2 i x^)^
/^ fix Va^ ± x^ 1 /^ <^x
148
xWa-±x- ^^^ ^ la-J X
-^a? ± x"
C ffx Vx^ — a^ , 1 , /x\
J x'^xJ' — o? 2a-x2 2^3 \^ay
160. I , = o Vx'^ ± a^ ^ - log (x + Vx^ ± a^).
151. I = — o Vct^ — X- + 77 sm !-•
, ^„ . c?x Vx^io^
152.
J /• rfx
^ x^Vx^ zh a.'
2 a^x
183. f- ■'^ "'""-"
x'
154
^ = + log (x + Vx» ± a^.
X- X a
156. f / = ,--^ 4-log(x+V^^T7^.
X^(?X X . , 35
, - — = ■ , — sm-i-'
• (See Note on pages 20-«l.)
IRRATIONAL ALGEBRAIC FUNCTIONS. 23
air \ du
158. Cfj^^ = ,jCf(
g'^ — cu^ j (^^ — cu^^ '
where u —
159. I , - = - \ f( 1 du, where u^ = a -^ cx^.
J Va + ex'' cJ ' \ c J
D. — Expressions Involving Va -\- bx + cx^.
Let X = a -\- bx + cx^, q = 4: ac — b^, and k = In order
q
to rationalize the functio n f(x, Vft + bx + cx^) we may put
Va + ^x + ex' = V± c V^ + Bx ± x^, according as c is posi-
tive or negative, and then substitute for x a new variable z,
such that
z = ^A + Bx + a;2 ± a-, if c > 0.
V^ + Bx — X' — vCi a ^
z = ■} if c < and >• 0.
X — c
= \- -f where a and /? are the roots of the equation
^ a — X
a
A -\- Bx - x^ = 0, if c < and <
— c
By rationalization, or by the aid of reduction formulas, may
be obtained the values of the following integrals :
160. C^ = ^-logfVX-\-xVc + -^\iic>0.
^ -\fx Vc \ 2Vcy
/dx —1 /2c.r + i>\ 1 . , ,/2c.>' + h\
24 IRRATIONAL ALGEBRAIC FUNCTIONS.
' -^ X^fX q^/X
r dx _ 2(2cx + b)Vx 2k {71- 1) r dx
^^" J X" VX ~ (2 n - 1) ^A'« "•" 2 w - 1 J X«-i Vx'
67. Cx^VXdx
^ {2cx + b)^X f hX , 15\ 5 r dx
12c \ '^ 4k Sk'J Wk^J VX
68. rx-vx..^ (^-+^)f:^^ + ,;"V, r^.
J 4(7i + l)c 2(% + l)A;J Vx
/xdx^_ Vx b r dx
VX~ c '2 c J Vx
dx
Vx"
69
/* a;c?cc _ 2 (^ij; + 2 «)
■ J xVx ~ ^Vx
/ xdx Vx A r ^^
X"VX~ (2ri- l)oX«~2"J J^VX'
"• J vf = (ro - jp) ^ + -87- J vx •
' J xVx c^Vx c J Vx'
IRRATIONAL ALGEBRAIC FUNCTIONS.
25
174
Vn
x^dx
x^Vx
175
{2b''-4:ac)x + 2ab 4 ac + (2 n - 3 ) 5' P dx
{2n-l)cqX'^-^Vx (2n-l)cq J X^-^Vx
x^dx
■f
Vx
'x^ 5bx 5b^ 2a'
3 c 12 c2 "^ 8 c3 3 c2
X +
'3 ab 5 b^
4 c^ 16 &
)f
dx
Vx'
176. fxVXdx = ^^^^~~ f-yXdx.
J 3 c 2 cJ
177. fxXVxdx = ^V-^ - ^ fxVx dx.
J 5 c 2 cJ
xX^dx X'^VX
P
S
X^dx
Vx (2*i + l)c 2cJ Vx
179. Cx^VXdx =(x-^}j
180
5 b\ X^X 5P-4ac
4:C
16 c2
/Vx
c?a;.
r x'^X'^dx _ xX'^Vx _ (2w + 3)^' r xX'^dx
' J Vx ~2(?i4-l)c 4(7i4-l)cJ Vx
^— r
2 (71 + 1) c J
181. Cx^Vxdx = (x^-
X^lx
(n + l)cJ Vx"
8 c 48 c' 3cy 5 c
'3ab
+ 1
8c2
32cvJ
VXrf;
'JCt
-^ xVX Va \ a; 2Va/
26
IRRATIONAL ALGEBRAIC FUNCTIONS.
183. I — = — ^sm-M -.= ] , or — = smh ^ ^•
184. f ^
dx
2VX
— , ' II a = 0.
ox
185.
d
xX"
Vx
Vx
(2 n - 1) aX
1 r dx b_r
" aJ -rX^'-'^^fx 2aJ
dx
186
■/
dx
Vx
a;'VX
dx
x«Vx
«« 2aJ a;VX
188
»^ X
X
189. fj^^iKl^
Vx (2 ?i - 1) Vx
Vx
'/
X"-''dx . ^> rx^'-^dx
X
Vx
iC^
a?
+ « I ^ +
dx
.xVX -' Vx
U--
VJ
2-' xVx ^
dx
190
Vx'
191,
r x'^dx _ 1 r x'^-^d,x _ h_ C x-^-^dx a Cx'^'
' ^ x»Vx c J X'^-WX cJ A'«VX cJ x»
f
■-r"'X"6ga; _ x"'~'^X"Vx _ (2n + 2m-l)h P x^^-'^X^dx
Vx ~ {2n + m)c 2c(2n + m) J VX
{vi -\)a r x'^-^X'^dx
{2n + 7n)cJ Vx
192,
r dx
Vx
(2n+27n-3)l>
(w — 1 ) ax'" - ^X " 2(t{m~l)
?i + m —
{in — 1) «
'> r dx_
'Vx
(2 ?^ + m - 2) c Z'
rf.r
x
m-2^Y«VX'
IRRATIONAL ALGEBRAIC FUNCTIONS. 27
■
r X"dx _ _ x"-'Vx (2?i-i)^> r X"-'(ir
(2 7^ - 1) r X''~^dx
m. - 1 J ^-'"--'VX
194. r/(a-, V(a; - a) (x - ^')) fZx
/\hu^ — a u (b — a) [ u du
where u^(x —b) = x — a.
E. — Expressions Involving Products o f Powers of
(a' + b'x) AND Va + bx + cx\
Let X = a + bx + cx^, v = a' + b'x, q = 4:ac — b^,
/3=-bb' -2 a'c, k = ab'^ - a'bb' + c«'^ then
195. I — — = —j=log "^ —
yVX VA;
tan"^
V- k 2V^-kX
= sin~^ 7=="' iiA;p£0.
V— ^ b'v^—q
„ r dx 2 b' Vx . „ , „
196. — p= = 7. ' if ^ = :
^ v^X
(So
r dx _ ,_ ■> /^ + 1 ,
/- _^ ^ _ &'Vx _ _^ r dx
■J r^Vx" ^'^ 2aJ vVx
■Jy^VX 3/3i;^ 3/3J ^;VX
28 IRRATIONAL ALGEBRAIC FUNCTIONS.
200 f ^dx ^ 2(2k+Jv)
««, rvdx b'^^ B r dx
201. -7= = TT I -1^'
202. Cv^Xdx = ^-^^^-i~^Vxdx.
r vdx __ _ b'Vx _ ^ r dx
J X" Vr " ~ (2 .. - 1) cX» 2 J A'« VX ■
r^^x^ _ &'x»Vx _ _^ rxv^
J VX ~ {2n + l)c~ 2cJ VX '
r (Za: _ _ bWx _ (2m-3)f3 r dx
■ J ^'«VX (m — 1) A-y™-' 2(m — 1)A-J ^,m- iVX
_ (m-2)c p dx _ ^ ifyt-z£0.
(m — l)A-c/ ym-aVx
206
1/ y;
,y-«Vx (2 m -1)^^;"
_ 2(m-l)c f _^^_ ■ . , _ rt
{2m-l)(^J ,.»-iVx'
r Vx^x _ _ ^>'XVX _ (2m-5)/3 T Vx^
J t;-" " (m-l)A;v"'-i 2 (m - 1) ^- J t-""-!
(m - 4) c rVxdx
~ (m- l)kJ v"'-^
~(w-l)^*'2(^ *'"'-■ ^^J i;'"-'Vx"^ ''J i;"'-2Vx^
_ 1 /_ &'VX _ r dx _ J r rfa; \
~ (m - 2)6'='(^ v—' V v'-Vx ^^J v-'-'VxJ
IRRATIONAL ALGEBRAIC FUNCTIONS. 29
208. (v'-<Xdx=- — \-—-{vv^-^X^'x
J {in-\- L)c\
- {m+ ^) ^C v'"-'^^ dx- (m-l)kC v"'-'-Vx dx\
^^« f* dx
209. 1=
1 / h'Vx , , , ^^^ c dx
+ (m + 2n-2)c f ^^ ^ Y if A;5z£0.
■ J i'- .Y« VX ~ {2m + 2n-l)li\ v"' X»
+ (m + 2?i-l)c f ^^-T=Y if A; = 0.
* r X"dT
211. I ^L^^,
1=
v^'-WxJ
+ (2n-l)kf-
,, ^ r X''-^dx\
A'»-ic?a;
1 / &'X"-Vx ..^f x»-
30 IRRATIONAL ALGEBRAIC FUNCTIONS.
213
Vx {m-\-2n)c\
v"" (Zx 1 fVv^-^ Vx
'v^'-^X'^dx'
/V"
Xn
Vx (m — 2 ?i) c \ A'"
^ '^^J X«VX ^ ^ -^ X"VaJ
+
{x + a) (x + Z*) VX (S - a){x + a) Vx (a — b)(x + b)^X
1
Va + bx -\- cx^ ± Va' + b'x + r'j:-
_ Va + 6a; + (?a;^ ip Va' + b'x + c'.
a — a' + ((^ — ^') a; + (c — c') x^
Vx Vx Vx
a;2
(x + a)(a; + 6) (^' — a)(a; + a) (a — Z») (cc + J)
(a- + a)VX _^rf^^ (a-Z^)Vx
X + 6 x + b
C lax' + Z* _ . „. . .
c/ \ ' -2 j_ i i^^ IS ^^ elliptic integral.
/cc Va + bx^ , 1 r ,
; dx = — I Wab' — a'b + by^ ■ dy,
Va' + b'x"" bWb'^ ^ ^ ^>
' x Va + ^a;'-^
where y^ = a' + b'x\
MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 31
IV. MISCELLANEOUS ALGEBRAIC EXPRESSIONS.
V2 ax — x^ •dx = — ^ — ^sr2ax — x^ -[- — sin"^
a
«,^ r dx . . X 1 /-, ^\
215. I — - = versm"^ - = cos~^( 1 )
^ V2 ax - x2 ct \ */
^-« . a;"c?a; a,w-iV2acc — x^
216.
V2aa;-
^2 71
a (I -2 n) C x^'-'^dx
/
n J V2 aa; - x^
/
217
*^ a;''V2 ax — aj^
+
dx a/2 ax — x^
x»V2ax-x2 a{\-2n)x-
n — 1 C dx
1) a J rpU—l
(2 w - 1) aJ a;«-i V2 ax - x^
//- 5 , x"-iV(2ax — x^)"
X" V2 ax -x^-dx= ^^-— r ^
71 + 2
_^ (2.^ + l)a r„_,V2^^3T..,
w + 2 J
r V2 ax - x^ • 6?x _ V(2 ax - x'')^
71 — o /' V2 ax — x^- c?x
— 3 /' V2ax -c
(2 71 - 3) a J x»^
220. I — , = — sec-^ ( - )•
32 MISCELLANEOUS ALGEBRAIC EXPRESSIONS.
221
•/
dx 1 , Va^ + x^ — a
, = — log ,
a;Vx"-|-a2 an Va^ + a;" + a
222
223
/x?dx
%■ sin~l / —
Sin
CC"'
a
/
• (Za;
(a + ^'a;2)Vx hV"^
log I
'a; + 8' + V28^'
+ tan-M 1 +
'2x
-tan-M 1
\
'2x
where iS* = a.
c^.dx 1 r^ ,/^ , V2x\ ^ y. V2.t\
a + 6ic^
— log ( , — ) )■ , where hV' ^ f-
V a + bx?
225.
^ -dx 2 V«
/ x^ -dx _ 2Va; a T
c?a;
(a + hx?) Vx
226
c?a;
Vac
+
r
dx
227.
■-^ (a + ^-x2)=2V^ 2a(a + Z»a;2) ' 4 a J (« +7»a;2) V^
/^ ^sfx-dx x^ . 1 r ^Jx-dx
+
(a 4- ^'a;'-')^ 2. a{a -\- hx^ 4aJ (a + bx^)
iaJ (a
If tti, ttj? «8> etc., are the roots of the equation
Pffic^ +^ia;"-i +x>2^''~'^ + • • • +i9„ = 0.
the integrand in the expression
/
{P(fic" + piX"~^ + • • • +p„) Va + 6x + cx^
MISCELLANEOUS ALGEBRAIC EXPRESSIONS. 33
where m<in, may be expressed as the sum of a number of
partial fractions of the form , ^==i and these
{x — af?f^ a -\-hx-\r cx^
can be integrated by the aid of equations given above. Thus,
228. f (P- + 9)dx
J (x- a') (x - b') Va 4
+ bx -\- cj?
_q -\- a'p P dx
~ a'-b' J (x-a')Va
229
■ f-
-' (a'
(x — a') Vo. -\- bx + cx^
q + b'p f* dx
a'-b' J (x - b') -^a-\-bx + cx^
dx
(a' + c'x") VoT
cx^
^ f an-1 ^ ^ («C' - «'«)
tan""^ X
-\/a'{ac' — a'c) ^ a' (a + cx"^
1 , Va '( a 4- ea-- ) + .r Va 'c — nr'
log ^
230
2 Va'(a'c — ac') ^a'{a + ^a'"^)- x Va'c — ac'
(a' + c'a;^) Va + cx^
1 .,..-, |o'fa + ex')
Vc' (a'c — ac') ^ a'c — ac'
1 1 Vc' (a 4- ex*) — Vac' — a'c
2Vc'(ac'-a'c) " Vc'(a + cx^) + Vac' - a"c
231. \fU, \/^n^[^^^
where z"(a' + 6'x) = a -\- bx.
z"~'^dz
34 MISCELLANEOUS ALGEBRAIC EXPRESSIONS.
232. ffi^, V c + V'a + bx) dx
where «" = c + Va + bx.
where y* (a' + b'x) = a -\- bx and s is the least common multi-
ple of n, q, etc.
234. ff(x, ■Va + bx + a;2) dx
r f2Va-z-b s^Va-bz+Va \ (z^Va - bz +Va)dz
where xz + Va = Va -\- bx + x\
235. ("/(a;, Va + ^»cc + x"^) dx
- C A '^^~^ V? - bu -V aVl{bu - a - ti^) du
~J'^\b-2u' 2u-b J {b-2uf '
where u = Va + bx + x^
x.
'x2+axV2 + aA „^ _,/aa;V2'
-'a;^ + a* 4a8V2l W-ax V2+W
r_rf^ = J_ f lo, f ^::^«A _ 2 tan- Y^^ I
a^— x^.
TRANSCENDENTAL FUNCTIONS. 35
V. TRANSCENDENTAL FUNCTIONS.
236. j sin x -/(cos x)dx = — \ /(cos x) d cos x.
237. j cos X -/(sin x) (/a; = | /(sin a-) rZ sin a;.
238. rsinx-/(siua;, cos x)dx-^- \f{^^ - «^ «)<'^>
where z — cos x.
239 r__^^-_ = _l— / f-^-f-^l,
*^''- J a^h cos a; e(^ - (') \J z ■\- c J z-c j
where s = tan ^x, and c- = (^^ + «)/(^ - a). [See 651.]
, , . = 1 , ^; "^, 5' where s; = tan i a;.
(X ± (^ sm a; »/ a ± 2 fts; + az'
241. ^/(sin a-) 6?x = -ffUos (^| - ^) ) '^ (^| " ^J'
242. r/(tan a-) rfa; = - J /ctn f | - xj d (^| " ^ j'
243. j"/(sec a;) dx = - J/csc f | - a- J fZ (^| ^ ^ j'
rsmx^f(sui^x)dx^ r J^z) dz
J Vl-A;^sin2x ^ 2 V(r-^^) (1 - k^^z)
where z = sin'^a;.
r cos.r./(cos^x).7.r ^ r.A l-^)^^ . ^here z = sin^x.
J Vl-A-^sin^a; -^ 2 V^ (1 - A;^ «)
36 TRANSCENDENTAL FUNCTIONS.
/' tan X -/(tan^ x) dx _ C J z \ dz^
where 2: = sin^ aj.
247. r/(«x + ^)<^^ = ^ ^ f{ax + S)c?(aa^ + ^').
248. rsec" + 2a; ./(tan cr) (Za; = (^(l + z^yf{z) dz; « = tan 3
249. I /(sin x, cos x) dx
= — I /( cos I — — X], ?,\n\ — — x]]d\ — — x
|_^Uin(|-.))c/f|
//• <i (x) dx
fix) ■ sin-i x-dx = sin-^ x ■ ^ (a;) — I ^^ ^ _^ > t/ic,
where <f>(x) = i f(x)dx.
/C <b (x^ dx
f(x)-cos-^xdx = cos-^x-<f,(x)+j y ' ^ -
252. \f{x) ■ t2,n-^xdx = tan-^ x-<f,(x) - j t^ f '
253. r/(a;) • ctn-^ ic dx = ctn-^ a; • <^ (a;) + f^M^ .
254. j /(a;, cos x)dx = — I ./"( o ~ «> sin z j c?2:,
where 2; = — — x.
Li
/' sin a; -/(cos x)dx _ IT J z — a\ dz
J a-\-bcosx bJ \ b J z
where z = a + b cos x.
TRANSCENDENTAL FUNCTIONS. 87
256. ff(x, ^ogx)dx^jf(6', z)e'dz, where « = log x.
257_ r fQogx)dx ^ Cf{£^dz, where z = log x.
258. Cx"'f(log x) dx = Je('" + i>y(s) dz.
259. (/(sinar, cos a;, tan x, etna;, sec a:, csca;)c?«
' -J-^Vr+T^' 1 + ^^' 1-^=^' Iz'l-z^' 2z )
2dz , ^ ^
■ ) where z = tan - ;
1 + z' 2
VT^^2 1 1'
:» where z = sinaj;
Vi - «'
= I / , > z, -y VI + s^
: J where z = tan a; :
l-\-z^
://(vi, vi^, Vr^^, W' vfe' 7:
:> where « = sm-'a;:
2^z(l-z)
J''\^i + z vi + v^ ^ ^ y
2V2(1 +«)
:j where z = tan^x.
38 TRANSCENDENTAL FUNCTIONS.
260. I s\nxdx= — cos x. [See 247.]
261. I sin^ a; c?x = — ^ cos a; sin a; + 1^ cr = ^ a:; — :|^ sin 2 a;.
262. I sin^ xdx = — -^ cos x (sin^ x + 2).
«o« r • „ 7 sin"-^a; cosa; . n — 1 f . „ „ .
263. I sm''xdx = 1 I sin«-^a;c?a;.
J n n J
264. I cos a3 c?x = sin X. [See 247.]
265. I cos^ xdx = ^ sin a; cos x -'c \x = ^x + \ sin 2 aj.
266. I cos^ a; c?a; = -J sin a: (cos^ x + 2).
267. I cos" a; c?a; = - cos" ~^ a; sin a; H | cos"~^a;c?a;.
J n n J
268. I sin x cos xdx = \ sin^aj.
269. I sin^ x cos'' a; c?a; = — ^ (:|^ sin 4 a; — x).
270. I sin X cos"» a; c?a; = —
cos'^ + ^as
i.p
271. I sin"'a; cos a;c?a; =
in + 1
sin"» + ^a;
m+ 1
272. j cos"" a; sin"x(Za; =
'•/
273. I cos"'x sin"a;rfx =
cos'^^^a; sin^ + ^a;
rth -\- n
m — 1
m + w.
sin"~^a; cos'^+^a:
4- — — r— I cos'""'' a; sin" a; (/a;.
m + w.
+ "^ . ^ I cos'"a; sin"-''a:rfx.
TRANSCENDENTAL FUNCTIONS. 39
, ri^^m^xdx 1 / sin"-\'r , ^ ^ rsiii"-^a-rf.»'\
274. I ■ = \ — \-(n — 1) I
J cos™ a; n — m \ cos'"~*a; ^ J cos"* a; /
1 /siu^ + ^-c , , _, rsin"a7(fe\
= 7 \ hi — m + Z) I r—
m — 1 \cos'"-^a; ^ ^J cos"-'*x/
_ 1 / sin"-^a; rsin«-2xrfx\
~ m — 1 \cos"'^^a; ^'^ ^J gos"'-^xJ'
rcos,^xdx _ cos^' + ^a- m — ?i + 2 /'cos'"a:;c?cc
"J siii"a: (/i — 1) sin""-'^ n — \ J sin"~^a;
'^x m— 1 /^cos'""'^.'
)in"^^a; m — ?^c/ sin":
J_ r cos'"-^xd.
n — lsm"~^x n — lJ sin"~^a;
(m — n) sin" ^x m — nJ sin" a;
1 cos'"~^a; m — 1 /*cos"'~^xdx
. . , _cos™ [tz— x]d[— — x
, sm^xdx r V2
/sin'«a:;c?a;_ r
cos"ic »/
sin" ( 77 — a^
277. I -^ = log tan x.
J sin x cos x
278. ^i- = log tan - + - )
J cos a; sin^a; \4 2/
— CSC X.
279. j
sin^'x cos" a;
1 1 ,m + w — 2/* dx
m-\-n — 2 r _
3"~^x n — 1 J sii
n — 1 sin"*" ^ a; -cos"" ^x n — 1 J sin"'a; -cos^^^a;
1 1 m + n — 2 r dx
~ m — 1 sin"*~'a; •cos"~^x m — 1 J sin"*"~^a; -cos".-?;
^ r dx 1 cos a; , m — 2 /* dx
280. I -. = r -: \ T I • ^ , •
40 TRANSCENDENTAL FUNCTIONS.
dx 1 sin X . n — 2 C dx
C dx 1 sm a- ii — 2 r dx
J cos"a; n — 1 cos'^^-'a; n — \J cos"~^
282. I tan xdx = — log cos x. [See 247.]
283. I iaxi^xdx = tan x — x.
284. I tan"a;fZic = r^ — j tan^-^xc^ic.
285. I ctn xdx — log sin x. [See 247.]
286. I ctn^xdx = — etna; — x.
287. jctn^aic^a; = - ^ "_ J^ — jctn"-^xdx.
o.^^ /^ T -1 i /''■ , »'\ ,1 1+sina;
288. I sec xdx = log tan I - + - 1 = |- log -t—
289. I sec^a;c?a; = tan x.
sin a:
/r dx sma- n — 2C «^
sec^a^rfx = I = ■; TT \ — I I -^^
c/ cos"a; (72, — 1) cos" ~^ a; w — l^cos"~^a;
sin a; , n — 2 C , ,
= 7 ;^ r~l 1 7 I sec"~^a;ria;.
(n — 1) cos"~^a: n — lJ
291. I esc a; cifx = log tan -^ X.
292. I csc'^xc^a; = — ctn x.
TRANSCENDENTAL FUNCTIONS. 4-1
293. I csc"a;rfx = | -r-^
cos X , n — 2 r dx
n — 2r dx
"-^a; n — 1 J sin"-^i
(n — l)sm"~^x n — lJ sin"~"^ic
cos x n — 2 f „ ,
= — 7 ..s . „ 1 1 7 I csc^-^xdx.
(n — 1) sm"~^a; n — lJ
294. ftr-r^^ = - tan (i tt - i x). [See 241.1
J 1+ sill X V4 -J / L J
/dx
: = ctn a TT - ^ x) = tan (i tt + ^ a;).
1 — sin a; ^
296. 1 :; = tan A x, or esc x — ctn cc.
»/ 1 + cos X
/dx
= — ctn 4 a-, or — ctn x — esc x,
1 — cos X
298. f— -^^ = ^-^^ • tan- > (sec ^ • tan i a- ± tan 0),
if a > i, and b = a sin 6.
^„„ /* c?.x ± sec a , sin h(a ±x')
299. I r-^ = 7 — log f-) (»
J a ± sm a^ o cos -^(x ^ a)
if 6 > a, and a = 6 sin a. [See 241.]
««« /* dx — 1 . , r^ + a cos a;~l
300. I z = , • sm-' — —
J a -\- b cos X Va^ — b^ \a-\-b cos a; J
5
CO ?
" 5
o s
-Sim
QJ o o
•^ 5-
1 . , r Va'^ — b^ ■ sin a;~|
, sm- — :
I Q^ — ip- \_ a -\- b cos X J
b g or : sm- ' I r^ I ■>
or ,
_ii 1 - ,rVft2-^
£s-§ or
c o
« •a
Va* - y
tan~M y^ tan \ x V>
, r Vft^— ^ • sin a;~|
tan-M — T— h
L c» + a cos X J
42 TRANSCENDENTAL FUNCTIONS.
1 , r ^* + <^ COS a; + V6^ — d^ ■ sin x ~\
' >2 - a2 °^ L a + 6 cos a; J '
V^
or
1 , r V^ + ft + V6 — ft • tan ^ a; ~|
'-ft' LVZ* + ft-V6-a-tanlxJ'
1 , , , r V^»2 - ft2 .
or
, tanh-T \^^-"^-^^^^ l
V^2 _ ^2 ^ 6 _|_ (J cos aj J
_-, r dx 1 ^, ,
301. I — TTT = o , ,i, [b log (ft cos cc + ^> sm x) + ftxl
J a + 6 tan x a^ + ¥^ ^ /'J
/f/a; 1
^ \ = -7= log tan (i cc + i tt).
r _^mxdx_ _ _ /* COS {\ TT — X) d (^ TT — x)
' J a -\- b cos x J a + b sin (^ tt — a;)
= — - log (a + b cos a;).
304
/(ft' + J' cos x) dx _ b'x a'b — ah' r
a + b cos X b b J i
305
/
+ b cos X b b J a-\- b cos x
(ft' + b' cos a;) c?a; _ ab' — a'b sin x
(a + b cos x)^ a^ — h^ a + b cos a;
ftft' — bb' C dx
, ftft' - bb' C dx _„ „,.^
"• 2 TT I — TT [See 241.]
ft'' — 6^ c/ ft + ^ cos a; ■- -"
gQg r (ft' + ^''cosa;)c?x ^ 1 V jab' -a'b)^ix\x
■J (ft + ^ cos a;)" (w- 1) (ft2- Z*^) |_(a + />cosa-)"-i
r r(r^^<' - ^>Z'') (» - 1) + (n - 2) (ft/>' - a'b) cos a-] r/x l
J (ft + 6cosa;)"-i J"
307./
TRANSCENDENTAL FUNCTIONS. 43
(a' + b' COS x) dx _ (a' — h') tan ^ x
(1 + cos x)» ~ (2 w - 1) (1 + cos x)"-^
n (a' + h') - g.' r dx
2n-l J (1+ cos x)"-i '
308 C——^^— = - r - ^ sin a;
J (a + b cos xf (n - 1) (a^ - ^^^^ [_(a + Z» cos a;)"-i
+ (2n-3)aj ^^_^^ 'cos x)— ~ ^'' ~^U (a + b cos a;)'-^ J
309 f ^^ ^ tanja;
* J (1 + cos x)" (2 ?i - 1) (1 + cos x)"-^
^27i-lJ (l + cosx)»-^ L'e^-'-i^-J
otn r (*' + ^' COS x) f/a; a'b — ab' , , j .
310. I ^ — — — '- — - = — ^ -^ log (a-\-b cos a^)
»/ sin x(a + b cos x) a^ — ¥ ^ ^
H — ^ log sin ^x log cos -A- x.
a + b ^ ^ a — b ^ ^
»■... r ((t^' + b' cos x) dx a' ^ , ,, , .
311. I -^^ — —7-^ — - = -logtan|-a7r + a;)
J cos x(a + b cos x) a ^ ^
(ab'-a'b) f dx
- a'b) r
a J
a -\- b cos X
«,« r (a' -\- b' cos x) dx 4- («' =F ^') , i / i 7im
312. \ ,., ^ = ± ,\ ^ +i(a'db&')logtan|;
J sm a; ^1 ± cos a;) 1 ± cos a; '' ^ '^ * ^
313
/dx _ — ctn I" X
(1 - cos xy ~ (2 71 - 1) (1 - COS a;)
n — 1 C dx
n — l
4.iL:Ll.r_ ^h-—. ^See241.]
44 TRAMSCENDENTAL FUNCTIONS.
dx C dx
314
r dx r
ci^ — b"^ cos?x J (a^ — U^
1 . sin (a — x)
log ^^ :
2 ab sin a sin (a -\- x)'
~l "■ — 3"t^^~M~^^ )' whei
a^ sm /? \sm pj
1 a
or -s -: — — tan ' i -: — - j , wnere cos a
cos /3 b
„,^ /* tZcc 1 ,/&tanic
315. I p^ 5 . ,„ . „ =— tan-^i
a^ cos^ a; + ^^ sin^ a; ab
a
316. r44^ = ^^ tan- ftan . . J-^")
J « + b co&^x b^a \ ^ a + bj
X
~b
<«<» C sm a; cos a:" f/a* 1 , , , , • o s
J a cos- a; + 6 sura; 2(b — a\ ^ '
318. f ^ = C d{x-a)
J {a -\-b cos X + c sin a.')" J [a + ?• cos (a- — a)]"
where 6 = r cos a and c = r sin a.
319. I : [See page 61.]
J C '
dx
a -\- b cos X -\- c sin x
-1
sin"
J r i^ + c^ 4- a (b cos a; + c sin x)
^a^ — W—c^ |_ V(&^ + G^){a + ^ cos a; + c sin a;)
, ^= • log
V&2 + e^ _ ^2
]
[^^ + 0^4- a (& cos a- + csin a;) + V&^ + c^— ^^(isinx— ccosa;)"!
V(6^ + c^) (a + i cos a: + c sin a;) J
1 , V^-2 4- ,.2 - a2 _ c _j_ /^ _ (j) tan ^a;
OP
o
' X
■sIV + c- -a" ■\/b'' + c'-d' + c-(b- a) tan \
c^ [_ Va^ -b^-c^ J"
Va^ - 62
TRANSCENDENTAL FUNCTIONS. 45
/dx 1
— -— — : = - log (a -h c tan |a;)
a (1 + cos x)-\- c %\nx 'C
321 f ^^
J (a [1 + COS a;] + c sin x)^
1 r c (a sin a; — c cos x) i / , . i x ~l
= -o —^ r-;' ■ a log (a + c tan ix) •
^r.r. r (a; + sin a?) c^cc ^ ,
322. I ^— r^ — = a;tan-^a:.
J 1 + cos x
= |- sin a; Vl — P sin^a; + — sin~^(A; sin a;).
324. I sin x Vl — A;^ sin^ a; t?a;
= — -|- cos a; Vl — A;^ sin- x ^-j— log (k cos a; + Vl — A;^ sin^a;),
325. J sin a: (1 - A;^ sin^ x)^ dx = - i cos a; (1 - A;^ sin^ x)^
+ f (1 - k^) jsin X Vl - k"" siu^x dx.
««„ /^ cos a;c?a; 1 . , ,, . .
326. I , . ^= = T sin-i (A; sin x),
^ Vl — k^ sin^x ^
or - log (6 sin x + Vl + 6^sin^a;), where b^ = — k^
327. I- = — -log(A:cosx + Vl — A-^sin^a;),
•^ Vl — k^ sin'^ a; ^
1 . , Jcosx „ 2
or — TSin~' , , > where 6. = — k^
b Vl+62
^„„ /• tan a; c?a;
328. - =
^ Vl — A;^sin^a;
1 , / Vl - A;=^ sin^a; + Vl - k''\
;log
2Vi^^ \ Vl - A;2 sin^a; - Vr^A:^
46 TRANSCENDENTAL FUNCTIONS
xdx
/v cix
-;——. = - a; tan |a TT - a;) + 2 log cos i (i tt - x).
1 + sin X ^
/cc doc
' : = X ctn -^ (^ TT — x) + 2 log sin i (i tt — x).
J. olU X
331. iz = a; tan 4- cc + 2 log cos -J- a;.
J 1 + cos X
332. I :: = — x ctn ix + 2 log sin 4- x.
J 1 — cos a;
««« r tan a; fZa; 1 .( ^h — a \
333. , ==— =^=cos-M p— -cosa;)-
^ Vet + ^ tan^ X -\/b — a \ V6 /
334. r-^^^^- = ^-yU-\^--tan-»( J-.tana- ) .
J a -\-b tan^ a; a — b\_ ^a \* /J
««,. r tana^c^x
335. I
a + h tan x
= — — \bx — a log {a + b tan x) + a log sec a; V
336. I a; sin a;fZa; = sin x — x cos x.
337. I a;^ sin xdx = 2x sin a; — (a;^ — 2) cos x.
338. I a;^ sin a; c?a; = (3 a;^ — 6) sin x — (x^ — 6x) cos x.
339. J x"" sin xdx = — x™ cos x + m % x™~^cos xc?x.
340. I X cos xdx = cos x + x sin x.
341. I x^ cos X (fx = 2 03 cos x + (x^ — 2) sin x.
342. I x' cos X cZx = (3 x'' — 6) cos x + (x^ — 6 x) sin «.
TRANSCENDENTAL FUNCTIONS. 47
343. I iC" COS xdx = x'" sin x — in i x'"~^sin xdx.
.... rsincc , 1 sin ic , 1 Tcos cc ,
344. I dx = ■ —-—J H 7 I ——-7 dx.
J x"* m — 1 x'"-^ vi — lJx'"-^
«.., rcosx , 1 cos a; 1 rsincc ,
345. I dx = T ■ T T I 7«a;.
J X'" m - 1 a;'«-^ m - 1 J ic'"-*
346. J c?a; = a; - 77-77^ + ^^^ - ^^^7-, +
' x^
X 3-3! '5-5! 7-7! 9-9!
,6 a.8
«>.« rcos X , , x^ , X* a;" ,
3^^- J -^^^' = ^°^^-2:2!-^4T4!-6T6l + 8.8!
rxdx _ . x^ 7x^ 31a;^ 127 a-«
^^^- J sinx"'^'^3.3!"^3-5-5!"^3-7-7!"^3-5-9!"^
r^^ _ ^ g;^ 5.T^ 61^« 1385 a:^"
J cosa;~ 2 "*"4-2!"^6-4!'^8-6!"^ 10-8! '
/x dx
. „ = — a; ctn a; + log sin x.
sin'' x
351. I — 5— = a; tan X + log cos a;.
J COS'^X
362. n^ I aj^'sin^xtfo;
= a;'"~^ sin"""^ x (m sin x — nx cos a;)
' + w(?i — 1) rx^sin^-^XfZx — 7;i(w — 1) ra;"'-2sin"a;c?x.
353, 71^ J x"' CDS'* xc^a:
_ a;TO-i cos"~^ X (m cos x + nx sin x)
4- w(w - 1) fx'" cos'^-'^xcZx — m(?» - 1) Cx'"-'^ cos"xdx.
48 TRANSCENDENTAL FUNCTIONS.
354. f^
1 r a:!"*"^ (m sin x -{- (n — 2)x cos x)
^ (?i - 1) (?i - 2) |_ sin«-ia;
365. f^
»/ COS" a;
1 r x^~^(m cos g; — (?i — 2)a; sin a:)
^ (?i - 1) (w - 2) L cos»-ix
^ 'J cos"~^x ^ ^J COS" -^ a; J
^^„ /'sin" re c?a;
356. J
X'"
X ((m — 2) sin x -\- nx cos a;)
™m— 1
1 r sin"-^
~ (m - 1) (7n - 2) L
- n'j —;;;=r- + "(« - 1) J ^..-^ J
«^-. . 'cos" a:c?a;
357
1 P cos"~^ a; (na; cos a; — (m — 2) cos x)
^ (m - 1) (??i - 2) L ^'""^
„ /*cos"xfZx , . ., ^cos"~^xdx~\
— ^ I 5 \- n(n — 1) I 7, — •
J a;'"-^^ ^ V a;'"-'' J
358. I x'' sin"* a: cos"a:c?x
= a:^"' sin™ a: cos"~^a;(» cosa; -j-(')/i + n)x sin a;)
(in + 7i)'' |_ \i \
+ (n — 1) (//I + ?i) I X'' sin^x cos"~^a:(^x
TRANSCENDENTAL FUNCTIONS. 49
— m'p I a;''"^ sin^^^a; cos"~^a;c?ic
— p(^ — 1) I x^''"^ ^v^'x Q.o^'^xdx •
= •; — ; — -5 a:^""^sin"'~^a;cos"a:;(» sin a; — (w + w^a; cosa?)
(m + ny |_ v^ \ / /
+ (m — 1) (m + 11) I aj^ sin"*"- a; cos" a;c?a;
-{-np i xP~^ sin"'~^x cos'^'^xdx
„-n C ■ ■ 7 sin Cm — n) a; sin(m + w)a; ^
359. I sin mx sin nx ax = — ^r-^ i ^77 — ; — r~ *
J Z(m — n) Z{7n -\- 71)
__- /* . - COS (m — n)x cos(m + n)x g,
360. I sin mx cos nxdx — tt, r —rr) ; — . ' "
J 2(m — n) 2 (771 + 71) '^
»
03
-_- /* , sin (m — w) a; , sin(m + w)a;
361. I cos waj cos wa; rfa; = — T-7 r 1 777 ; — f— •
,/ 2 (m — 7i) 2(m + 71) •-'
362. I sin^ mxdx = tt — ("ma; - sin mx cos ma;).
»/ 27n^ '
363. I cos"^ mxdx = pr — ("mx + sin mx cos mx).
./ 2 ??i ^ '^
1
364. I sin mx cos mxdx = — -f^ cos 2 mx.
J 4m
365. I sin nx sin"'x<Zx = — ; — — cos nx sin^'x
J m + 71 1_
+ m I cos(/4 — l)x -sin^'^xc^x
50 TRANSCENDENTAL FUNCTIONS.
366. I sin nx Gos"'xdx = ; — — cos nx cos™ a;
J m + n [_
+ ml sm(7i — l)x-cos'''~^xdx \.
367. I cosnxsiu^xdx = ■ — sinna; sin'"a;
J m + n |_
-«/sin(„-l)x.sin"-..<^.].
368. I cos nx co^'^xdx = ; — sin nx cesser
J m + n\_
+ m I cos (w — 1) a; • cos"'~'icc?a; •
369
/cos nxdx _ /^cos {n — V)x dx /^ cos(w — 2)xdx
cos"'a; "" J cos"'~^a; J cos"'a7
rcosnxdx __ ^ r sin(?i — l)a;cZa; /* cos (?^ — 2)a-c?g;
' J sink's; ~ J sin'"~ia; J sin'" a;
/'sin wa;fZa; _ ^ /^ cos(?i — l)a;c?a; r sin(??- — 2)a;6?a;
' J sin"'a; J sin'"~^a; J sin"'a;
/'sin ??,a'fZa; _ ^ /' sin (?i — l)xdx /' sin (n — 2)a;c?a;
" J cos'" a; »/ cos'"" 'a; J cos'" a;
^P + n-lfl^
/'(cos ;:»a; + i s,\nj)x)dx _ . /';s''
■ J cos wa; »/ 1 + ;5;^" '
where z = cos a; + t sin x. This yields two real integrals.
- . /'('cos wa; + i sin «a-)f7a; ^ rzP'^"~'^dz
374. I ^ ^— ^ = ~ 2 I — — -,
^ sm na; J 1 — z^"'
where z = cos a; + i sin x. This yields two real integrals.
TRANSCENDENTAL FUNCTIONS. 51
r(icosx — smx)dx_ r dy
375. J -J 2—;!'
where y = z^zzm This yields two real integrals.
V cos nx
X
«-,, r ■ • , • 7 , rcos(a — 6 + c)
376. I sm ax sm ox sin cxax= — x i ^ ; — ;
J \^ a — b + c
cos (b + c — a)x cos (a + b — c)x cos (a -\-b + c)x \
6 + c — a a + 6 — c a + 6 + c J
'inn C z. J , fsin (a + ^> + c)
377. I cos ax cos te cos ca; aa; = i i ^^ — —-. — ;
J y_ a-\-b -\- c
sin (6 + c — ft) a? sin (rt — 6 + c) a? , sin (a + b — c)x \
b ■\- c — a a — b -\- c a + b — c j
cos (a -^ b -h c)x
X
378. I sin aa? cos bx cos ca; c?a; = — :f i
cos (6 -f c — g) a! cos (a -\- b — c)x cos (a + c — i) x
}
b + c — a a + b — c a + c — b
««« r -7 • 7 , r sin (a, -F ^» — c) a;
379. I cos ax sm bx sm ca;ax = f s ^^ — --. —
J I a -\- b — c
sin (a — b -h c)x sin (a + b -\- c)x sin (Z* -f- c — a.) a;
a — 6 + c a -^ b -\- c b + c — a
380. I sin~^a;c?a; = x sin~"^a; + Vl — x"^.
381. J COS'^XC/X = X QQS~^X — Vl — x^.
382. I tan-^xc/'a; = x tan~^x — \ log(l + x^).
383. j ctn- ^xdx = x ctrr ^ x -}- -^ log (1 + x^).
x\
62 TRANSCENDENTAL FUNCTIONS.
384. I s.eor'^xdx = x sec~^x — log(x + Vcc^ — 1).
385. I csc~^ a;c?x = x csc~'a; + log(a; + ■\f^— 1).
386. I versin-^ icci^a; = {x — 1) versin"' x + V2 a; —
387. C {?.m-^xfdx = a; (sin- ^ a;)^ - 2a; + 2 Vl - a;^ sin- 'a;.
388. I (cos-^ a;)^c?a: = x (cos"^ a;)^ — 2 x — 2 Vl — x^ cos~^ x.
389. fa; sin-^x^x - i[(2x2 - l)sin-ix + x Vl - x"].
390. I X cos~^xc?x = i[(2x^ — l)cos~'x — xVl — x^].
391. I X tan""'xc?x = ^[(x^ + l)tan~'x — x].
392. I X ctn-'xc?x = ^[(x" + l)ctn-'x + x].
393. I X sec~'xc?x = ^^[x^ sec""^x — Vx^ — 1].
394. I X csc-^xc?x = ^[x^ csc~'x +a^'x'^ — 1].
395. I x"sin-'xc?x = — — r ( x" + ^ sin-^x - \ , ^^ \
J n-\-l\ -^ Vl - xV
396. I x"cos-'xcZx = — -— I x" + icos-'x 4- \ ^ ,
J n + \\ J Vl -x^
i
■^>-ck^ ^ ^---e'"'
^^'■rc^
TRANSCENDENTAL FUNCTIONS. 53
397. Jx"tsin-^xdx = ——(x'' + HEin-^x-Cj
398. Cx»ctn-'xdx = —^(x" + 'ctn-'x-^ f^!^l^Y
X / iC
400
tan~^a;c?a; , .,.-..„. tan~^a;
^^ = logo; - 1 log(l + x^) -
X
401. Ce'''=dx = —- Cf{e"^)dx=J'^^^^^^, y^e'^.
a;e«^rfa; = — (ace - 1).
x^e'^'dx = I x'^-'^e"^dx.
a a J
P^ax 1 r e"^ , c^'^d^~\
404. I — dx — : + a I '
J x"" m — 1(_ x""-^ J a:"'-ij
J log a (log ay (log a)^
n(n-l)(n-2)- • '2.1a^
~ (loga)" + i
a
bx
-«« Ta^'^a; 1 r «■" a^-
407. I — — = T --r - 7
log a
2)a;»-2
a^ • (log ay
+
(w-2)(ri-3)a;"-3 (w
(logq)"-^ r a='dx ~\
- 2) (n- 3) "-2.1 J x J"
,^„ Ca^dx , , , , (cc log a)^ , (a; log a)
3
+
J
54 TRANSCENDENTAL FUNCTIONS.
409. I :: = Ior i ^-^^ ' ■*-
J 1 + e^ '' 1 + e^
/dx 1
411. \ —z;z-^-, — -: = ^=tan-M e^-^A/Tr
412. )— ^^== j=\\og(-\/a + be""'-Va)
/ /— 2 -\/n 4- /)^'"^
- log (Va +6 e"- + V^) L or == tan- ' ,Z_1 ^
VI -y/ZTa V- a
J {\+xf 1 +x J a(n + 1)
A-iA C ^^ 7 e'^ia sin ?;.x — » cos px)
414. I e"" sm »a? (/a? = — ^^ -^ f ^-^•
Atn C r,^ 7 ^'^ (f' cos px + » sin »a^)
415. I e"" cos »x- dx = — ^^ ^, — ^ ^-^ •
416
e*" log a; c^a; = 2 I
a aJ X
goa: gij^2 xdx = J— — ^ ( sln X (o, B\Xi X — 2 cos cc) + - ] '
/e'" / 2\
goa; cos'^ajc^a; = — — — - ( cos a; (2 sin x + a cos a;) + - 1 •
419. I e'^sin"6xc?x = -5— — ;-^( (a sin bx
J a^ + tr¥ \ ^
— nb cos bx) e'^ sin"~^ Jx + n (n — l)b- i e^ sin"~^ bxdx
TRANSCENDENTAL FUNCTIONS.
55 .
). I e'" cos" bxdx = -^—, — ^-^ ( (a cos bx
J a^ -\- n-h' \ ^
+ nh siu te) e"^ cos"~^ hx -\- n {n — X)h^ I e°^ cos"~26icc?a; )•
421. re^^tan^ajc^ic
n
/'
e'^tan"-ixc?a; — | e'^tan"-2a;c?x
422. re''^ctn«a;(Za;
e'^ctn"-!^
423
/
n-1
e"^ dx
+
a
n
-J
-/'
e'^ctn''-^xdx— | e'" ctn"-^ a; c?a;.
a sin X -\-(n — 2) cos a?
pOX V i
sin" X {n — 1) {n — 2) sin''"^ x
+
a^ + (71 - 2y re^^dx
(n - 1) (7^ - 2)
-2) J sin"-^
x
424.
/
e"^dx
^ a cos X —(n — 2) sin a;
cos" a; "^ (n — 1) {n — 2) cos""~^aj
a^ + (71 - 2)^ r e'^'dx
+
-2^ f
-2) J (
(» — 1) (ti — 2)»/ cos""~^a;
426. I e"^ sin"' a; cos"xdx
= 1 ; — TT"^ — ^ 1 ^""^ sin"^ X cos"~^ x (a cos x + (m + n) sin a;)
(w + ny + a^ K. ^ \ / /
— 7na I e'^sin"'~^a; cos"'~^a;(^a;
+ (?z. — 1) (m + n) I e"^ sin"' a; cos"~'^a;c^a; V
66 TRANSCENDENTAL FUNCTIONS.
= ~, ; — r^~. — :, \ e"^ sin"*"^ x cos" x (a sin x — (m -\- n) cosx')
(m + ny -\- a^ {. ^ ^ ' '
■\- na \ e'^sin'"~^x cos"~^icc?a;
+ (m — 1) (to + ii) I e"^ sin"*"- x cos" xc^x \
= •:^ 7^; -c\ re"^cos"""^a;sin"'~^a;('asina;cosic + ?isin^a;
(m + ny -\- a V
— mcos^a;)] + 7t(w — 1) | e"^ sin'"cccos"~^icc?a;
-\- mim — V) I e"^ sin'"~^ X cos^iccZo; [-
= -A re"'^sin"'~^£CCOs"~"^a;(asinxcosa:+ wsin^x
— m cos^x)] + w(w. — 1) I e"^ sin'"~^iccos"~^a;<Za;
+ (m — n) (m + n — 1) j e"^ sin"'"^ x cos" xc?x V
= -^\ f(e''*sin"'~^a;cos"~^a;(asinxcosa; + nsin^x
{m + ny + a^ [^ ^
— mcos^a;)] + m(m — 1) j e^^sin^^^xcos^^-xc^x
— (m — w) (w + w — 1) I e'"sin'"x cos"~^xo?x V .
426. I log xdx = X log X — X.
427. rx'"logx(Zx = x"' + ir^^^-- — ^^--^1-
428. j (log xydx = X (log x)" — 7i j (log x)"-^ c?x.
/x'""'' Vlos xV 7? /*
x™ (log x)" fZx = \ V T I a;'"(logx)"-'rfx.
^ ^ m. 4-1 m + l»/
TRANSCENDENTAL FUNCTIONS. 57
(log xydx _ (loga;)""'"^
430. f - ^.
J X /i + 1
^3lJl|^-log(log.)^Iog.^(^V(|ifV
432 f ^^^ = ^ + _^f^
J (log xy {n - 1) (log xy-^ n-lJ (log a;)"-'
r x"'dx a:"' + ' w +1 T a;"'c?a;
J (iog^~~(»i-l)(logx)»-i ?i-lJ (ioga:)"-i"
■- — ^ = I dy, where ?/ = — (m + 1) log x.
log a; J V
435. r ^^ -\o.(\o<rx) and f C^^-D^--^ ^ -^
J xlo^x- ^°^ ^^°^ ''^' J X (log X)" (log a;)»-i
436. flog («2 + cc2) (^a- = a; ■ log {a^ + x'')-2x + 2a- tan-' (^ ^
437. j (a + Ja?)'" log a;6?a;
438. j a;"' log {a + to) o?a;
1 r /'a^'" "•" ' dx~\
= ^^M^T r"" ^"^^" ^ ^"^ ~ ^ J "^Tto J *
/^log (g + ^3^) dx
439
X
, to 1 fbx\ , 1 /toV
= loga.logx + --2i(^-J +3i(^-; -
68
TRANSCENDENTAL FUNCTIONS.
r \ogxdx
**"• J (a + bxy
441
_^ 1 r log a; r dx "I
6 (m — 1) |_ (a + te)"'-i J cc (a + te)"'-ij
/loa: xdx 1 , , , , ^ 1 /*log (a + ^ic) dx
— ^ = ^logx.log(« + fa)-J X
442. I {a + bx)\ogxdx=- — log a; ^-f ax — :^to^
443. I ,^
= - (log a; — 2)Va + bx +Va log(Va + 6x -fVa)
— Va log ( Va + 6a; — Va) , if a >
= - [(log a; - 2)Va + 6a; + 2V^ tan" ^ ^^^^^^ 1 .if a < 0.
444. I sin log xdx = ^x [sin log a; — cos log a^].
445. I cos log a;c?a; = \x [sin log x + cos log a;].
446. I sinh xdx = cosh x.
447. I cosh xdx = sinh a;.
448. I tanh a; c?a; = log cosh a;.
449. I ctnh xdx = log sinh x.
TRANSCEIJTDENTAL FUNCTIONS. 59
450, I secli xdx = 2 tan~ ^ e'.
451. I csch a;c?a; = log tanh -•
/I . % — 1 /*
sinh"x(Zx = -sinh"~^a;-cosh a; I siuh"^^ xc?a;
sinh" + ^ a; cosh cc — r I sinh" + ^ a; ^x.
7i + 1 w +
/I . w — 1 /*
cosli"xc?a; = -sinha:- cosh""' a; H I cosh"~2a;^x
w n J
= sinh a; cosh" + ' a; H -^ | cosh" + ^ a; c?a;.
n + 1 n-^lJ
454. I x sinh xdx = x cosh a; — sinh x.
455. I a: cosh xdx = x sinh a? — cosh x.
456. j a;'' sinh xdx = (x' + 2) cosh a; — 2 a; sinh x.
457. I a;" sinh xdx = x" cosh a; — wa;""^ sinh a;
+ n(n — 1) I a;"~^ sinh xdx.
458. I sinh^ a; c?a; = |^ (sinh x cosh x — x).
459. I sinh a; • cosh xdx = \ cosh (2 a).
460. I cosh^ a;(Za; = | (sinh x cosh a; + a;).
461. I tanh^a:c?x = a; — tanh x.
60 TRANSCENDENTAL FUNCTIONS.
462. I ctnh^ xdx = x — ctnh x.
463. j sech^ xdx = tanh x.
464. j cscli^ic c?a; = — ctnh x.
465. I sinh~^ xdx = x sinh~' x — Vl + x^.
466. I cosh~^ a;c?a; = x cosh~^ x — Va- — 1.
467. j tanh- ^ cc (Za; = x tanh"^ a; + |- log (1 — a;^).
468. Cx sinh-i rrt/cc = ^[(2 x^ + 1) sinh"^ x - cc Vl + x^].
469. I ic cosh-^ a;6?a; = :J[(2a;^ — l)cosh-^ x — xVx^ — 1].
* ./ cosh a
+ cosh a;
= csch a [log cosh ^ (a; + a) — log cosh ^ (a; — a)].
= 2 csch a • tanh-^ (tanh ^ a- • tanh ^ a).
/dx
; -. — = 2 CSC a ■ tan- ^ (tanh 4- a; • tan i^ a).
cos a + cosh X \ ^ ^ /
/dx
- — \ ; — = 2 CSC a • tanh" ^ (tanh ^ x ■ tan A a).
1 + COS a ■ cosh X \ i ^ /
473. j sinh x ■ cos x c?a; = ^ (cosh x ■ cos a; + sinh x ■ sin x).
474. I cosh X • cos xdx = ^ (sinh a; • cos x + cosh a; • sin x).
475. j sinh x- sin a;c?a; = ^ (cosh a; • sin x — sinh x • cos x).
TRANSCENDEiSTTAL FUNCTIONS. 61
476. I cosh X • sin xdx = \ (sinh x • sin x — cosh x ■ cos a).
477. I sinh (ma;) sinh (wa;) c?a;
= — ^ 7, m sinh (?ia;) cosh (pix) — n cosh (jix) sinh (mo;) •
478. I cosh (mx) sinh (wa;) dx
= — ^ J m sinh (na;) sinh (wa;) — n cosh (wa;) cosh (vix) •
479. I cosh (mx) cosh (raa;) 6?x
= - 2 _ — ^ m sinh (ma;) cosh (wa;) — n sinh (/ia;) cosh (mx) ■
/ ■ dx _ r (Z(tana-)
a cos''^ X -\- c sin x • cos x -\- b sin^ a; J a + c tan a; + 6 tan"'^ ar
/(I + 7/1. cos a; + w sin x) dx _ T (m cos 8 + ?i sin 8) cos s • dz
a -{- b cos a; + (^ sin x J Z
' I ■ dz r (7)1 sin 8 — n cos 8) sin s ■ dz
+ . , . ^
where b — q • cos 8, c = q- sin 8, s = a; — 8, Z = a. + y • cos 2;.
C . , , X • V , 7^ 7 [See 303 and 304.1
I sm (mx + a) • sm (na: + 0) aa; ^ -^
sin [?«a; — nx + a — &] sin [?/ia; + '^^ + <* + ^]
~ 2 (??i — 7i) 2 (?/i + «)
I cos (mx + a) • cos (iix + i) rfa;
sin [^mx -{- nx -\- a -\- b"^ sin [mx — nx -\- a — b"]
^ 2 (wi + n) ^ 2 (??i - n)
I sin (mx -\- a) ■ cos (nx + h) dx
cos [ma- + wa^ + «■ + ^] cos [^mx — nx + a — b~\
~ 2 {m + w) 2 (?/i — w)
62 MISCELLANEOUS DEFINITE INTEGRALS.
VI. MISCELLANEOUS DEFINITE INTEGRALS*
481. J x''-'^e-''dx= j log- dx = T(n).
T(z + l)=z-T(z), if z>0.
r(jj)-T(l-y)=^yiil>y>0. r(2)=r(l)=l.
r (w + 1) = n I, if n is an integer. T (z)= Il(z — 1).
r(i) = Vt^. Z(y) = i)^[log r(t/)]. Z(l) = - 0.577216.
>,oo C w^ X ,, r" a:'"-^^/:^- T(m)T(n)
482. a:'"-'(l -a;)"-irfa;= I -— — — — = ^\ , .-^ •
483. I sin"a;c?a;= ) cos"xdx
%/o «/o
484.
1-3-5 •• -(71-1) IT .^ . . ,
= o A r> — . s -77' if w IS an even integer,
J • 4 • D • • ' in) Ii
= ■ ^ _ ^j if 71. is an odd integer,
= \ Vtt — ^ {-•) for any value of n greater
rTl + lJ than-l.
J'^'^sinmxc?ic 7r.„ ^^^.„ ^ 7r.„ ^.
= -) if m>0; 0, if m = 0: — — > it ??i<0.
a; .^ J
* For very complete lists of definite integrals, see Bierens de Haan, Tables d'inti-
grales (Ufinies, Amsterdam, 1858-64, and Nouv. Tables d'intigrales difinies, Leyden,
1867.
MISCELLANEOUS DEFINITE INTEGRALS. 63
,^^ /^* sin x» COS ma;c?a; ^ .- ^ ^ ^ ^
485. I = 0, if w<- 1 or m>l;
»/0 X
— > ifw = — 1 or m = l: — > if — l<7n<l.
4 Z
.__ C^ &Ui^xdx TT
486. I ^ — = TT •
Jo x^ 2
487. J cos{x^)dx = J siii(a;2)(Zx = i\|--
sin A;a; • sin mx dx = \ cos A:a; • cos mx dx = 0,
*/o
if A; is different from m.
489. I sin^ mxdx= I cos^mccc^a; = jr*
.-- r'^ COS mxdx TT ^ ^^
»/o 1 + a;^ 2
.-, f"^ eosxdx C^ smxdx ir
491.1 ^z^= I -- = ^-.
492. r"e-"'^^x = ^V^- = j^ra).
c/o 2 a 2(z ^^-^
493. I a;"e-°^c?a;= ^ ^. ^ =^-7-
.«>. r" , » , 1-3-5- • •(2?i-l) Pr
Jo 2" + ia» ^a
e ^dx = ^ ^^ - a>0.
496. I e-"^ VxcZa; = 77- \/- •
«/o 2 n ^ 7i
497. f"^c^a; = V-- a>0,
Jo Vx ^ ^i
64 MISCELLANEOUS DEFINITE INTEGRALS.
dx IT
498
•f
499 r*__^^^_^
sinli (ma;) • sinh (nx) dx = \ cosh (ma;) • cosh (nx) dx
= 0, if m is different from n.
cosh^ (mx) dx = — \ sinh^ (mx) dx =
502. I sinh (mx) dx = 0.
cosh (mx) dx = 0.
sinh (mx) cosh (wx) c?a; = 0.
sinh (mx) cosh (mx) c?a; = 0.
506. I e~ "^ cos mx dx = -5— ; ? if a >■ 0.
a^ + m^
a-* + m^
J'' m
e-"^ sin mxdx = -r-; ;» if a > 0.
a-* + m"^
6-'^'=^ cos hxdx = -^ a>0.
Za
••X'l^
509. I ^:^^^c;a; = -^-
x o
510. rM£^=_^.
511, r'J<^,&=-^^
»/o 1 — a;'' 8
MISCELLANEOUS DEFINITE INTEGRALS. 65
1 + x\ dx ir^
512. f\og(l±^).^ =
»/o \1 — X/ X
513. r^i^l^ = -?log2.
Jo Vl-a;2 2
515. J (loga;)"cZa;=(- l)«-w!.
516.X'(lo.i)'.. = ^.
518. f , '^ = V?.
519. ra:"'log(-)(Za:= T'^''t,2i >^f^ + 1>0, ^ + 1>0.
520. flog ('?^V = T-
Jo ^ \e^ - ly 4
jr IT
log sin xdx = \ log cos a;c?aj = — — • log 2.
.0 c/o 2
X ■ log sin xdx = — — log 2.
523. I log (a±b cos x)dx = tt log ( ;r ] • « ^ 6.
66 ELLIPTIC INTEGRALS.
VII. ELLIPTIC INTEGRALS.
d6 r^ dz
where k^ <d, x = sin <fi.
E {4>, k)=f Vl - k^ sin^ e ■ dO.
y
(1 + ri sin^ ^) Vl - k' sin^ ^
<^ = am u, sin <^ = cc = sn m, cos <f> = Vl — aj^ = en m, tan (f> = tnu,
A<t> = Vl - A;2 sin^ <^ b Vl - kV = dn «, A;'^ = 1 - A^l
t< = am~^(<^, ^)=sn~^(a;, A;)=cn~^(Vl — x% k)
= dn-i(Vl-;k2^2, A;).
JS:=i^(i7r, ^), K'=F(i7r, k'), E=E{^'ir, k), E'-E^Tr, k').
T* 7 2 A;* sin 2 (0
It ko = q — — 7 and tan <^ =
1 + k k + cos 2 <i)
524. r ''
»/0
Vl - k^sin^e
= I [i + (i)'.' + [^) k' + (ifl)"^ +.,.], it ,.
<1.
= K.
525. Vl- k'sin'^ede
). J Vl
=i[-(«--(i^yf-G^yf--]""'<^-
ELLIPTIC INTEGRALS. 67
,474 X'O'O A -J c. I I
= J'\<i>, k),
3 5 5*3
where VI4 = i sin^ «^ -f- — , ^ = ^ sin* <^ + — sin^ <^ + ^^^,
A = isin«<^ + g^sin*<^ + g^sin^<A + 3^g^
. r* Vl - k^ sin' 6- dd = - <i>- E -\- Bva <!> cos, A ~k'
1
= E{4>, k).
J , = sn-i(aj, k)
V(l - a;2) (1 - k'^x')
528. . ,
V(l - x") (1 - k'^x')
= F{sh\-^x,k). Q<x<l.
529. r , '^"^ =cn-^(g, A;)
»^^ V(l - x^) (A;'2 + Ji-x^)
= F{cos-^x, k) = sn-i ( Vl - x% k). 0<x<l.
530. C , "^^ =dn-i(a;, A;)
= #(A-ia;, k) = sn-^ Q Vl - x% k\ < a; < 1.
531. r , ^"^ =tn-\x,k)
-'o V(l + X') (1 + A;'2a;2)
= i^(tan-'a;, A;) =sn-Y--=^=' A; Y < cc < 1.
\ Vl 4- cc^ y
* The next forty-two integrals are copied in order from a class-room list of Prof.
W. E. Byerly.
68 ELLIPTIC INTEGRALS.
532. r . ^^ =2sn-U^,k)
^0 -Vx (1 -x)(l- Jc'x) '
= 2F(sm-Wx,k). 0<a;<l.
533. r—==J^== = 2 cn-i ( V^, k)
^^ Vx (1 - x) (k'^ + k^'x) ^ ^
= 2 i^(cos-i V^, k) = 2 sn-i(Vl-cc, A;). < a; < 1.
534. r ^ ^^ = 2 dn-i ( V;, A;)
•^^ Vx (1 - a;) (x - k'^) ^ [
= 2i^(A-i V^, k) = 2 su-i Q Vl -x,k\- 0<a;<l.
535. r , "^^ =2tn-^(V^, /^)
^^'o V(l + a;) (1 + A;'2a;) ^ ^
= 2i?'(tan-iV^, A;) = 2sn-Y^-^,>tY 0<a:<l.
536. r , ^^ ^lsn-Yf,-^Y a>6>a;>0.
537. r , ^^ ^lsn-Y^,^Y ^>a>^.
538. .
^x V(a2 4- x") (b^ - x")
cn-if?, ^=i=Y b>x>0.
V^M^' V^ V
539. J^ ^^
J -a
ELLIPTIC INTEGRALS. 69
dx
V(a* - a;2) {x' - b^)
1 ,/ \a^ - x^ la'' - b^\
541. r ^^
'" V(x2 + ay(^M^')
542. X' ''^
V(a; - a) (x - y8) (a; - y) .
y
Va-y K^^-y ^a-y
V(a; — a) (cc — /3) (a- — y)
2
Va^
y
(V!^;- V!5^)
'^ V(a — x)(x — /3) (a; — y)
545,
Va — y \^«-^
c?a;
V(a — a;) (a; — )S) (x — y)
2 ^..-i/'./^Lziy i^^^. J^-/3^
Va-y V^«-/5^-y ^«-y.
c?a;
V(a^^^) (/? — a;) (a; — y)
2
Vo-"
a > a; > 6.
1^ ,/a; /a^ - b'\
a > /3 > y.
^sn-Ur^, V^^^V a.>a.
sn-M \ ^' \r^ ^ • x>a.
' --('^/^»■ Vfi|)- <'>->^-
sn-i(-V ^ ^' -V ^ )• a>a->^.
Va — y
546. f^
\ iS — y a — a; ^a — yj
70 ELLIPTIC INTEGRALS.
'^ dx
5«X
y V(a — x){fi — x) (x — y)
2
sn"
548
Va — y
■(^§i^. V!5^)- ^>^>y
^ V(a — cc) (/3 — cc) (y — a;)
Va
549. f
7
(V^.- >/^)
sn-^UL:^, X^^-^ • v>a;.
V(a-x)(^-a^)(y-a;)
2 , / /a — y
Va — V \ ^a — a;
sn-M \ ^' \ ^ • y>x
550.
X'
a > y8 > y > 8.
dx
V(x - a) (cc - /3) (a; - y) {x - 8)
2
, / 113-8 X -a l/3-y a-8\
V(a-y)08-8)
a;>- a,
531. f"- "^^
V(a. — a;) (a; — /3) (a:; — y) (x — 8)
\^a — j8x — 8 ^a —
^ y-8^
552
V(a - y) (/3 - 8) \^a--^x-8 ^a-y/3-8j
a>x> fS.
■X
^ V(a - a;) (a; - ^) (a; - y) (x - 8)
\^a— ^x— y ^ a
2 _g^-ir J«zi^y ^-^. J«^i^ 1^
V(a - y) ()8 - 8) \^a-^a;-y >'a-y^-8y
a > a; > /3.
553
X
ELLIPTIC INTEGRALS. 71
^ dx
^ -si {a - x) (/3 - X) {x -i){x- 8)
V^/3-y a-x' ^a-y ^-l)
2
sn"
V(a - y) (/? - 8)
^ > CC > y.
554. J''^ "^^
^ V(a - CI-) (^ - a;) (cc - y) {x - 8)
2
sn"
V^)8-y a;- 8 ^/a
;8-y o^^
■V(a -y)()8-8) V^^-y^-S ^a-yj8-8^
i8 > a; > y.
555. f^ "^^
V(a -x){(3- x) (y - a;) (x - 8)
.2 s,,-irj^^ y^^. J^^:il y-8
Va
V(a -y)(/3-8) V^y-S^-a; >'a-y^-8y
y > a > 8.
556. J^ ^^-^
's V(a - a;) ((8 - x) (y - x) (a; - 8)
\ *y — 6 a —X ^a
1 .^-if.l^nj^^^^^. J^l^lI y-g
V(a-y)(^-8) \^y-8a-x ^a-y(i-8j
y>x>8.
X^ (Ir
V(a - x){fi- x) (y - a;) (8 - a;)
2
sn"
i/^J^-y.g-^ /)8-y a-8\
V^a-8 y-a;' ^/a - y )8 - 8/
V(a-y)(/3-8)
8>a;.
558. i sna;c?x = - cosh~M -yp ]•
559. I en a; c?a; = - cos~^ (dn a;).
72 ELLIPTIC INTEGRALS.
560. I dn xdx = sin~^ (sn x) = am x.
561. r-^^^iogf" 'Y^ 1-
J s,nx [_cn X 4- dn a; J
^nn r (^^ It F^' SH 0? + dll X~\
562. I = - log •
J cnx k' \_ en ic J
_„„ /* c?aj _1 J FA;' sn X — en a?"]
' J dn X k' \_k' sn ic + en xj
sn^xdx = Ti[a^ — -E^(ania;, A;)],
565. j en^ajc^cc = — [^(ama;, A;) — k'^x"].
566. I dn^a;c?a; = ^(am x, k).
567. (m + 1) fsn'^ajc^x = (m + 2) (1 + A;^) fsn'^+^a^cfaj'
— (m + S)k^ I sn^ + ^a^c^a; + sn'" + 'a; en a; dna;,
568. {m + l)k'^Ccn"'xdx = (m + 2) (1 - 2 k^)Ccn"'+^xdx
+ (m + 3)A;M cn'"+ *a;c?a; — cn'"+^a; snxdnx,
569. (m + 1)^'2 rdn"»a;c?a; = (m + 2) (2 - k^)fdn'" + ''xdx
— (m + 3) j dn"*+*a;c?x + A;^dn"*+^a;snaena;,
Since sin2 ^ = _ _ _ (i _ A;2 • sm2 ^),
J ^2 sin2 6id& \ r^ dd 1 /^2 ^
VI -A;3sin2(? ^Vo Vl _ A;2sin2(? *Vo
TRIGONOMETKIC FUNCTIONS. 73
Vm. AUXILIARY FORMULAS.
A. — Trigonometric Functions.
570. tan a • ctn a = sin a • esc a = cos a ■ sec a = 1.
tan a = sin a -j- cos a, sec^ a = 1 + tan^ a,
csc^a = 1+ ctn^a, sin^ a + cos^ a = 1.
571. sin a = V 1 — cos^ a = 2 sin ^ a ■ cos ^ a = cos a • tan a
fe=Vi
1 tana /I — cos 2a 2tan4-a
Vl + ctn^a Vl+tan^a ^ 2 l + tan^^a
=v
gpp- ^ "1
= ctn ^a • (1 — cos a) = tan -^ a • (1 + cos a).
sec^a
572. cos a = Vl — sin^ a = = = =
Vl + tan^ a Vl + ctn^ a
-4
1 + cos 2 a 1 - tan^ ^a „ , . „ ,
~?^ = -. , ^ — rf — = cos^ia — sin^ia
2 1 + tan^ ^ a ^ ^
= 1—2 sin^ |- a = 2 cos^ ^ a — 1 = sin a • ctn a
sin 2
a _ /csc^ ^ ~ 1 _ ^^^ i ^ — ^^'^ "2" ^
a * csc^ a ctn 4- a + tan 4- a
-„n ^ sin a Vl — cos^ a sin 2 a
o7o. tan a =
Vl — sin^a cos a 1 + cos 2 a
1 — cos 2 a _ /l — cos 2 a _ 2 tan ^ a
sin 2 a ' 1 + cos 2 a 1 — tan^ ^ a
sec a _ 2 _ 2 ctn |^ a
esc a ctn \ a — tan ^ a ctn^ ^ a — 1
74
574.
TEIGONOMETEIC FUNCTIONS.
1 —
— a.
90° ± a.
180° ± a.
270° ± a.
360° ± a.
sin
— sin a
+ cos a
T sin a
— cos a
± sin a
cos
+ COS a
T sin a
— cos a
± sin a
+ COS a
tan
— tana
T etna
± tana
T etna
± tan a
ctn
— ctn a
T tana
± ctn a
T tana
± etna
sec
+ sec a
T CSC a
— sec a
± CSC a
+ sec a
CSC
— CSC a
+ sec a
T CSC a
— sec a
± CSC a
575.
0°.
30°.
45°.
60°.
90°.
120°.
135°.
150°.
180°.
sin
i
i^
iVs
1
iV3
iV2
i
COS
1
iVi
iV2
i
-i
-iV2
-IV3
-1
tan
1
V3
1
V3
00
-V3
— 1
1
V3
ctn
CO
V3
1
1
V3
1
V3
—1
-V3
00
sec
1
2
V3
V2
2
CO
-2
-V2
2
V3
-1
esc
CO
2
^y^
2
V3
1
2
V3
^
2
00
576. sin ^ a = V^(l — cos a).
577. cos ^ a = V^(l + cos a).
578. tan ^ a = ^—
cos a
cos a
sm a
+ cos a sin a 1 + cos a
579. sin 2a = 2 sin a cos a.
580. sin 3 a = 3 sin a — 4 sin^ a.
581. sin 4 a = 8 cos^ a • sin a — 4 cos a sin a.
TRIGONOMETRIC FUNCTIONS. 75
582. sin 5 a = 5 sin a — 20 sin^ a + 16 sin* a.
583. sin 6 a = 32 cos* a sin a — 32 cos^ a sin a + 6 cos a sin a.
584. cos 2a = cos^ a — sin^ a = 1 — 2 sin^ a = 2 cos^ a — 1.
585. cos 3 a = 4 cos^ a — 3 cos a.
586. cos 4 a = 8 cos* a — 8 cos^ a + 1.
587. cos 5 a = 16 cos* a — 20 cos^ a + 5 cos a.
588. cos 6 a = 32 cos^ a — 48 cos* a + 18 cos^ a — 1.
2 tan a
589. tan2a =
590. ctn2a =
1 — tan^ a
ctn2 a - 1
2 ctn a
591. sin (a±ft) = sin a • cos /? ± cos a • sin )8.
592. cos (a± fi) = cos a • cos yS =f sin a • sin ft.
..«« , ^N tan a ± tan 5
593. ta.n(a±ft) = - — ^•
^ '^^ 1 rp tan a ■ tan /3
..«.. , ^x ctn a • ctn )S rp 1
594. ctn(a±/5) = — ^ ^ Z, '
^ ^^ ctn a ± ctn /3
595. sin a zt sin )8 = 2 sin ^ (a =b /3) • cos i(a + iS).
596. cos a + cos /8 = 2 cos ^(a + /8) • cos |(a - /3).
597. cos a - cos /8 = - 2 sin |(a + /3) • sin \{a- ft).
sin ("a d= S)
598. tana±tan/3 = ^ ^•
cos a • cos ft
-~^ « sin (a ± ft)
599. ctn a ± ctn ;3 = ± ^-^^ — r^-
'^ sin a- sin /3
76 TRIGONOMETRIC FUNCTIONS.
_„_ sin a ± sin yS , , , _,
600. ■ ^ = tan i (a i S).
cos a + cos p ^ ^
sin a dz sin /3
601. ^ = — ctn i (a + S).
cos a — cos /8 i \ » /
„-_ sin g + sin ^ _ tan -|- (a + j8) ^
sin a — sin /3 tan ^ (a — j3)
603. sin2 a - sin^ ^ = sin (a -\- (3) ■ sin (a - (3).
604. cos' a - cos' ^ = - sin (a + /3) • sin (a - /3)c
605. cos' a — sin' /3 = cos (« + /3) • cos (a — /3).
606. sin xi = ^ i(e^ — e~^) — i sinh x.
607. cos xi = ^{e^ + e~^) = cosh x.
608. tan xi = -^^ — ; — i tanh x.
6^ + e~^
609. e^+ 2'' = e^ cos ?/ + ie^ sin ?/.
610. a^ + 2'' = a=^ cos (2/ • log a) + ta'^ sin {y • log a).
611. (cos zti- sin ^)" = cos nd ±i- sin n^.
612. sin a; = — i i(e" - e"^). ^
613. cos a; = I- (e^" + e"").
614. tan x = — i -z—. •
e-^ + 1
615. sin (x ± yi) = sin x cos yi ± cos x sin yi
= sin X cosh y ± * cos x sinh y.
616. cos {x ± yi) = cos X cos 3/1 q= sin x sin yt
= cos X cosh y 4= » sin x sinh y.
TRIGONOMETRY. 77
617.
In any plane triangle,
a b c
sin A sin B sin C
618. a'^ = h''^-c''-2bc(toQA.
„-_ a-\-b _ sin ^ + sin B _ tan ^ (A -\- B) _ ctn ^ C
' a — b sin .4 — sin B tan ^ (.4 — i?) tan ^ (.4 — 5)
620. sin^^=^^^ — ^^ — ^, where 2s = a + 6 + c.
621. cosM=^pZ«).
622. tani^^>~/^^^;^>
^ >' s (s — a)
623. Area = ^ ic sin ^ = Vs (s — a) (s — 6) (s — c).
In any spherical triangle,
-„. sin A sin B sin C
624. -^ = —. — - = —
sm a sin b sm c
625. cos a = cos 6 cos c + sin 5 sin c cos A.
626. — cos ^ = cos B cos C — sin B sin C cos a.
627. sin a ctn & = sin C ctn B + cos a cos C.
«r.o , i /sin s • sin (s — a)
628. cosi-^=-V • r, ■ -•
^ ^ sin b ■ sm c
««« . , A /sin (s — 6) • sin (s — c)
629. sinAJ = -V — '^ — ^I — ^^
^ ^ sm b ■ sm c
««o. , . /sin (s — ^) • sin (s — c)
630. ta.ulA = \ ^ i , . ^ '
' ^ sm s • sm (s — a)
78 TrwIGONOMETRY.
oQi 1 ^ j cos (S -B)- COS (S-C)
631. GOS^a=\ ^ : j- r-^-; ^-
» sin f» . sm f ;
sin B • sin C
„r,n ■ 1 — COS S-cos(S — A)
632. smia = V ■■ — „ • ^ ^•
■^ \ sin « sm n
CQQ 4- 1 * / — COSTS' -cos (.S — ^)
633. tania = V eos(^-^).cos(^'-C) -
2s = a-{-b-\-c. 2S=A + B + C:
634. cos \{A-\-B) = ^^—^ '- sm \ C. ■
coK 1/^ x)\ sin-|-(a + &) .
635. cos ^ (J — ^) = T-^^-r ^ sin ^ C.
sin -^^ c
636. siniM + 5) = 2^^^^^P^cosia
^ -^ cos ^ c
637. sin \{A — B) = V^ cos \ C.
638. tani(^ + B) = 55ll|^etaia
639. tan«^-^) = ?|±i|^ctnia
640. tan |(a + ^') = ^"^ t ^^ 7 f x tan i c.
^ ^ ^ cos |(^ + 5) ^
641. tan ^(^ - b) = ^!" t /^ 7 ^x tan ^c.
g^2 cos ^(a + b) ^ ctn|C _
cos ^(a — b) tan -^ (^ f ^)
ANTITRIGONOMETRIC FUNCTIONS. 79
In interpreting equations which involve logarithmic and
anti-trigonometric functions, it is necessary to remember that
these functions are multiple valued. To save space the
formulas on this page and the next are printed in con-
tracted form.
643. sin-^a; = cos~^ Vl — x^ = tan~^
X
— sec~^
= CSC
X
1 - = 2 sin-i [^ - i Vl - a;2]i
= i sin-i (2 X Vr=T2) ^ 2 tan-» \^ — ^ |
= -^tan-M ^--^^-J=i7r-cos ^a;
= 4" TT — sin~^ Vl — x^ = — sin~ ^ (— a;)
= ctn-i^^^^!— ^^ = (2w-f^)7r-nog(a;+Va;2-l)
= i TT + ^ sin-\2 a;2 - 1) = ^ cos-'(l - 2 x^).
VT^-^ 1
644. cos~^x = sin'^ Vl — x^ = tan~^ = sec""^ -
X X
= -^TT — sin~^a; = 2 cos"
= |-cos-i(2a;2-l)
■v^
= csc~^ — ■ = TT — COS" V— aj)
Vl - a;2 ^ ^
= ctn-i ~
Vl
x*
i log (a; -f Va;^ — 1) = tt — t log ( V^^ — 1 — a;).
80 ANTITRIGONOMETRIC FUNCTIONS.
645. tan~^cc = siri"^ — , = cos~^ , = h sin-^ -— — ;
X
= -J-TT — tan~^ -
x-
L 2 Vl + x2 J L 2 ViT^ J
= ^ Un- ,-1^, = 2 tan- [^ll+l^']
1 — X^ |_ X J
= — tan~^ c + tan~^ :; = — tan~^ (— x)
[_1 — ex J ^ '
= i * log TT^ — ; = i * log -.
646. sin~^ a; ± sin~^ y — sin"^ [cc Vl — if ±y Vl — a;*].
647. cos~^ X zb cos~^ y = cos~^ [xy if V(l — x^) (1 — ?/^)].
648. tan-i a; ± tan-^ v = tan-^ ^^J- i .
648. sin~^ a; ± cos~^ y = sin~ ^ [a:/y zt V(l — x^) (1 — ?/*)]
= COS" ^ [?/ Vl — x^ If a: Vl — y^].
650. tan- ^ x ± ctn-^ y - tau- ^ f^^^l = ctii- ' [^^1
651. log (x + yi) = i log (x^ + y^) + i tan"" ' (jj /x).
HYPERBOLIC B'UNCTIONS. 81
B. — Hyperbolic Functions.
652. sinh a; = ^ (e^ — e^^) = — sinh (— x) = — i sin (ix)
= (csch x)~^ — 2 tanh ^x -7-(l — tanh^ ^^)-
653. cosh X = ^ (e^ + e~^) = cosh (— a;) = cos (ix) — (sech ic)~^
= (1 + tanh^ ^x)^(l- tanh^ ^ x).
654. tanh a; = (e^ - e"^) -^ (e^ + e"^) = - tanh (- x)
= — i tan (tx) = (ctuh x)~^ — sinh a; -^ cosh a;.
655. cosh xi = cos a;. t*^ z^ X ^ ^»*^V
656. sinh a;* = ^ sin a;. r57 -^^iX ^ eA-A/^^X
657. cosh^a; — sinh^a; = 1.
658. 1 — tanh^x = sech^a?.
659. 1 — ctnh^a; = — csch'^a;.
660. sinh (x ±y) = sinh x ■ cosh y ± cosh a; • sinh y.
661. cosh (x ±y) = cosh a; • cosh ?/ ± sinh a; • sinh y.
662. tanh (a; zt ?/) = (tanh x ± tanh ?/) ^ (1 rb tanh x • tanh ?/).
663. sinh (2 a;) = 2 sinh a; cosh x.
664. cosh (2 a;) = cosh2a;-|-sinh^a; = 2 cosh^a: — l = l+2sinh^a;.
665. tanh (2 a;) = 2 tanh a; ^ (1 + tanh^a;);
666. sinh (i a;) = V^ (cosh a; - 1).
667. cosh (J- x) = Vi (cosh a; + 1).
668. tanh (i a;) = (cosh x — 1) -i- sinh x = sinh a; -f- (cosh a; + 1).
669. sinh x -\- sinh y = 2 sinh -^ (a; + ?/) • cosh ^ (x — y).
670. sinh x — sinh y = 2 cosh ^ (x + 2/) • sinh ^(x — y).
82 HYPERBOLIC FUNCTIONS.
671. cosh X + cosh y = 2 cosh ^ {x + y)- cosh ^(x — y).
672. cosh X — cosh y = 2, sinh ^ {x + y) ■ sinh ^{x — y)
673. d sinh a; = cosh x ■ dx.
674. c? cosh X = sinh a; • dx.
675. c^ tanh x = sech^ a; • dx.
676. 0? ctnh a; = — csch'^ x • c?a;.
677. d sech a; = — sech a; • tanh x • c?a;.
678. d csch a; = — csch x ■ ctnh a; • dx.
dx
679. sinh-'a; = log (a; +Va;2 + 1) = J"
Vx^ + l
= cosh~^ Va;^ + 1.
680. cosh- 1 X = log (a; + Vx^ - 1) = J
Va;2-1
= sinh-^ Vx^ — 1.
/rfa
^-3
x^
/rfx
dx
683. sech- ^x = log (^^ + yj^, _ 1^ = - J
684. csch'^x = log f- + yj^ + 1 ) = "X
X Vl — x^
dx
685. c? sinh-^x =
686. c? cosh~^x =
xVx^ + l
<^x
Vl+X^
dx
VV
687. dta,nh-^x =
HYPERBOLIC FUNCTIONS. 83
dx
688. dGtnh-^x = -
689. dseGh-^x = -
690. dcsGh-^x = -
1-x^
dx
x^-1
dx
ic Vl — x^
dx
X
V^+i
If m is an integer,
691. sinh (mTri) = 0.
692. cosh {miri) = cos mTT = (— 1)*".
693. tanh (mTri) = 0,
694. sinh (x + mTri) = (—!)'» sinh x.
695. cosh (x + mTri) = (— 1)™ cosh (x).
696. sinh (2 m + 1) ^ Tri = ^ sin (2 m + 1) ^ tt = ± i
697. cosh (2 m + 1) i T^^ = 0.
-rr ±X ] = i cosh CC.
799. cosh (— ±a;j=±:i sinh x.
700. sinh w = tan gd u.
701. cosh u = sec gd m.
702. tanh u = sin gd u.
703. tanh ^ m = tan i gd w.
704. u = log tan (i tt + ^ gd u). Tsec a; r/rr = r/d' '
a:.
84 ELLIPTIC FUNCTIONS.
Elliptic Functions.
dz r* dO
V(l - s;2) (1 - A"2^=^ Jo Vl-A-2siii2^
where A- <C 1, and x = sin <^, <^ is called the amplitude of u and
is written am (u, mod A;), or, more simply, am w; x = sin <fi = smc,
Vl — x^ = cos <^ = en w, Vl — A;^a;^ = A<^ = An r< = dn %,
Hence, am(0)=0, sn(0)=0, cn(0) = l, dn(0)=l,
am (— u) = — am u, sn (— u) = — sn u,
en (— u) = en u, dn (— u) — dn m.
705. sn2« + cn2« = l.
706. dn2« + /.-2sn2?f = 1.
707. dn^it - k-' cn'u = 1 - k' = k'^
2 sn ?^ • en tt ■ dn ?(
708. sn2u =
709. en 2 M =
1 — k' sn'* «
cn^ i( — sn^?/ • dn^ u _ 1 — 2 sii^ ?t + k^ sn*u
1 - A;2 sn* ?< ~ 1 - A;2 sn* le
_ 2 sn^ ?< ■ dn^ u _ 2 cn^ ?<
1 — k^ sn* u 1 — k^ sn* u
„, » - _ dn^ ri — k' su'^ u ■ cn'^ u 1—2 k^ sn^ ?i + A;^ sn* u
710. dn 2 w = :j — J = — -^^
1 ~ k- sn* i< 1 — A;'' sn* u
_ . 2 l^ ^v?u-Q,v?u __ 2 dn^ u
1 — k^ sn* M 1 — k^ sn* ?<
m\ 1 — en ?/ 1 — dn II. dn ii — en m
711. sn2(
712. cn^l
2 J l+dn« k\l + cnu) k'^ + dn u— k^ en u
n 4- en u k^ en v — k''^ + dn u
'u\ _ dn
+ dn u k^(l + en n)
7c'- (1 +enM)
A;'^ + dn w. — k^ en w
ELLIPTIC FUNCTIONS.
713. dirl-1- i + dui* ~ /l-2(l + cni*)
85
~~ /v'^ 4- dn « — ^'■^ en u
If, moreover, v = I — , = >
Jo V(l - z') (1 - /tV)
714. sn^ u — sn^ y = cn^ v — cn^ ii.
„.. ^ ^ sn ?6 ■ en V • dn v =h en m • sn y • dn w
715. sn ill ±v) — —. :-, 5
^ ^ 1 — k^ sn- u ■ sn'' v
„, _ , ^ en u • en y zp sn ?( • sn v ■ dn ?? • dn v
716. en (u ±v) = 7^ — 5 i
^ ^ 1 — A;'' sn- tc • sn'' y
= en it • en V rp sn u • sn y • dn (u ± v).
„,„ , . . dn «-dn y =p />;'^ sn ?i • sn y? -en 7/ • en V
717. dn {u ±v) = ::; —„ 7, s
^ ^ 1 — k^ sn^ u ■ sn'' v
= dn ?< • dn v zp /*;- sn ?< • sn ?; • en (w ± v).
„,„ , , tn ?f -dn f ± tn w-dn it
718. tn Cu±v) = - 7 , ^
^ ^ 1 ^tu u -tn V • dn u ■ an v
«,« , , s . V 2 sn it-en vdn V
719. sn (w + v) -\- sn (?t — v)= r; — 5 5— *
^ "^ ^ ^ 1 — k^ sn'' « • sn'' v
/ N 2 sn ?' • en u ■ dn ?t
720. sn (71 + y;) — sn (u — v) = r^ — 7. r~ '
^ ^ ^ -^ 1 — /v'' sn'' ?t • sn'' V
. . 2 en it • en ?'
cn(?t + w) + cn(it— y)= :j 1-5 3 ^•
^ ^ ^ ^ 1 — /r sn- u ■ sn'' v
„„ _ ^ . , . 2 sn ti, ■ sn ?> • dn u ■ dn i;
Tad. en (it + iM — en ni ~ v)= 77 — 7, ^
^ ^ ^ ^ 1 — A:'' sn'' it • sn'' v
_,-„ ■, / ^ 1 / N 2 dn ?t • dn i>
7^ J. dn (u -\- v)+ dn (it — v)= 7-; ; ^ •
86 ELLIPTIC FUNCTIONS.
_^ . , , , , , , 2k^ smi-snv -cnu-cnv
724. dn (ic + w) — dn (71 — v)= .. _ , .^
sn^ u ■ sn^ V
sn^ u — sn'^ V
725. sn(t. + ^).sn(^^-^;) = ^_^,^^,^^_^^,
-y
1 — A:^ sn^ w • sn* v
1 Fdn^ V -j- k'^ sn^ ?* • cn^ v ~|
A;^ |_ 1 — A;^ sn^ u • sn^ v J
726. en (u ■}- v) ■ en (w — v) = rr — i ^
en^ w + en^ v 1 _ -1 ^^^ ''^ ' ^^^ ^' + ^'^^ '^ ' ^^^^ ^
1 — A;^ sn^ tc-su^v 1 — A;'* sn^ w • sn'' v
727. dn (u + v)- dn (m — v)
_ 1 — A;^ sn^ ?i — A;'^ sn^ v -\- k^ sn^ ii ■ sn^ v
1 — k^ sn^ u ■ sn^ t;
dn^ n + dn^ t?
-1.
1
-A:^
sn^
tt • sn^ V
sn?<
• dn w
• en V ±
sn ?' • dn V •
en u
1
-A;^
sn^
u • sn^ V
en?*-
dnw-
cnz?-
dn
vqzk'^ snu
• snv
1 — A;^ sn^ u • sn^ v
„_ _ , ^ , . sn M • en w • dn v ± sn ?? ■ en v • dn w
728. sn (u ± v) on {u rp v)
729. sn (m db v) dn (w qr v)
730. en(^±z;)dn(^zF^) = -^^ ^ j^ — sn^'t^^sn^.;^
-n- .-^ , . x-ir^ , x-i (en V ± snw-dn w)^
732. sn (wt, A;) = i sn (w, A;') /en (m, A:').
733. en (wt, A;) = 1 /en (m, A;').
734. dn {ui, k) = dn (m, k')/cn(it, k').
bessel's functions. 87
D. — Bessel's Functions.
7oO. t/g (x) — 1 2^ "^ 2^ • 4^ 2^ ■ 4^ • 6^
736. iq, (a;) = ^0 (a;) • log a; + 2-2 - 2F:4r2 + 22.42.62
7i! A (- i)^-^.» + 2t
[a^.= l + i + i + ... + l/k:]
[When 11 is an integer
737 T I't\ — "V 5^ -^ [When ?i is an integ
• « I . t/„ ,^x; r (?i + 1) ^ 2" ■^'^^'■kl(n + k) ! ^19 may be used.]
738. lK(^)=Jn(x)-logx-^JX ^''~2^^.iy'''
739. According as n is or is not an integer, A ■ J^i^) + B ■ K„(x),
or A ■ J^(x) + B ■ J_ ,^(x) is a particular solution of Bessel's
equation, fp.^ ^ r?^ / w^
+ -■—+ 1--Az = 0.
740. f/Jg (a;) /riic = — J^ (cc) ; 6? [ic" ■ J"„ (x) ] /(7ic = x" ■J^_i (x) ,
if /i > I ; (^[a;-« ■ J,Xx)ydx = — x.-" ■ J"„+i(a'), if ?i > - 1
741. J,^_,(x) - J„^,(x) = 2 . dj„(x)/dx ;
2 n ■ J^(x) = X ■ J"„_i (x) + x- J„+i(x).
When x is large it is sometimes convenient to compute
approximate numerical values of J^ (x) by means of the semi-
convergent series.
.^xrt x / N 2 r„ r (271 + 1)77 1
742. ^„(^)=^— I^P^.cosj^^ ^^_^|
(4 n? - 1) (4 n" - 9)
^ . f(2n + l)7r
-.}].
(4 rv" - 1) (4 ?i^ - 9) (4 v? - 25) (4 rv" - 49)
^ 4 ! (8 .x)*
744 n - ^^'-^ _ (4^^-l)(4r.'^-9)(4n^-25)
'"~ 8x 3! (8^)^ "^
88
SERIES.
E. — Series and Products.
[The expression in brackets attached to an infinite series shows values
of the variable which lie within the interval of convergence. If a series
is convergent for all finite values of x, the expression [x^ < co] is used.]
745. (a + by = a" + na^'-'^b
, n(n — V) „,„ , , n\ a^~^b'' , ^,0^0-,
+ 2! "''+•• • + (,.- A)! ,!.! +• -•■[*<"■]
746. (<i-Sx)-' = iri+- + ^ + ^' + ---1- pV<a».]
Ct I Cv ct- (M I
747. (1 zb xy = l±nx-\- ^ ^^~ ^ x^
2!
n{n — V) {n — 2) x^ (± 1)^' n\ x^
" 3] H • • • + ■^^^TT^yr^ +
748. (l±a;)-» = l=F?^^ + ^^^^^Vr^^'
2!
[a;2<l.]
3! ■ "^ ^^-^ {n-iy.k\
749. (l±c.)i-l±ix-^.T^±|^a;^
[x^ < 1,
J
750. (l±a,)-l = l=pj^+l^,x«^l^a^
751. (l±.). = l±i.-y.'±if|^^
1-2.5.8 ,
3-6 -9 12 "■
lx'<-L]
SERIES.
89
1-4 , 1-4-7 ,
752. (l±a:)-§ = l + ix + g^x2rF3-^a;'
1-4. 7-10 , r "^1 1
H 7i cc* zp • • •• \x- < 1.1
^3 -6 -9 -12 ^ •- ^
, a;* 1.3a;« 1.3-5ic«
753. (i±^¥ = i±i«^^-2:4±2:4:6~2T:6:8-'"-
[a;2<l.]
, 1-3 , 1.3-5 , ,
754. (l±^')-^ = l + i^' + ^^ +27176 "^^ ■■■
[x" < 1.]
755. (1 ± x)-' = l^x + x^-px'' + x*zpx'-\ . [a;2 < 1.]
31 , 311 3
756. (l±x)^ = l±ix-h^^x-:+j-^-^x'
3-1-1-3 3.1.1-3.5 ,<.^-j
+ 2.4.6-8 ^2.4.6.8.10 ^ •- ^
757. (l±^)-? = l + l^ + 2T4^'^t7I76^'"^"*' t^""'^^-^
758. (I±x)-^ = 1^2x + 3x''zp4:x^ + 5x*^6x^+ ' • :
Ix' < 1.]
759. e- = l+x + |-' + |-j + ---. [a;=^<co.]
Cicloga)^ , (x log ay , r 2^.^ n
760. a- = 1 + X log a 4- '^ ^° ^ + ^—o, + ' " •• [^'<^^-]
761. i(«^ + e-^)=H-| + fi + fi + - •• [^^<^-]
762. i(e^-0 = a^ + fj + fj + f-! + '' ^- [x2<=^.]
763. «-" = l-^^ + fi-fi + li • I^^<oo.]
90 SERIES.
A series of numbers, B^, B^, B^ •• •, of odd and even
orders, which appear in the developments of many functions,
may be computed by means of the equations,
2 n (2 n - 1)
2!
-^2n o, J^2n — 2
2n(27z-l)(2n-2)(2n-3) _ _
—^ ^-B,„_,=(2n-1)B,,_,
_ (2.-l)(2.-2)(2.-3) ^^^_^^...^_^^„_,^^^
Whence B, = h^2 = 1, B, = ^l, b, = 5,B, = ^\, B^ = 61,
B, = ^L, B, = 1385, B, = /^, Ao = 50521, Bn = ^Wo. ^12 =
2702765, Bis = i, etc. The ^'s of odd orders are called
Bernoulli's jSTumbers ; those of even orders, Euler's Numbers.
What are here denoted by -Z>2n-i ^nd Bz,, are sometimes rep-
resented by B„ and £J„, respectively,
7? 9
'2n — 1
(2?i)! (22«-l)7r2«
(2n
02n + 2r -I 1 1 1
7fi4 ^ _^ rr Ax^ B,x* B,x' B,x'
e^-1 2 2! 4! 6! 8!
[a;<2 7r.]
765. log X = (x - 1) - i(x - ly + i(x - ly .
[2>a;>0.]
[^>i-]
SERIES. 91
[a^>0.]
768. log(l-{-x) = x-ix^ + ix'-ix* + -- -. [a;2<l.]
769. \ogCj^^ = 2[x + ix'-b^x' + \x'+-- 'I [rr2<l.]
771. log(a;+Vl+a;^) = a;-— + ^^j7g- ^^ g^ + -- •.
[a;2<l.]
Series for denary and other logarithms can be obtained
from the foregoing developments by aid of the equations,
log„a; = log, a: • log„e, log.x = log„a; • log, a,
log, (—z) = (2n + 1) iri + log,s.
772. sin £c = a; - |-j + |j - li H . [x^<co,]
jp2 ^4 ^6
773. cosa; = l— 77T + — — TTiH = 1 — versinic. [x^ < oo.]
2! 4! 6!
774. tan a; - a; + g + ^g + g^g + 2835
02n/02»i 1\ J> rp^n — l
+ •••+ (2.)!"" +•••• [^^<i-^-]
^py_ J^ «// JO ^ (/^ t//
775. ctna; = ----^- — -^^
a;(2?i)! •- -■
92 SEEIES.
776. sec. = l + - + — +— +.- + -^^ + -.|_.^<-J
777. csc. = j + - + 3-g-, + —
ftmo • -1 i"'^ L • o X i. ■ o • O X
778. sm>a. = x + - + 2:^.- + 2:^.-
+ ■ • • = ^ TT — cos~^a;. [a;^< 1.]
779. tan-^cc = x — ^x"^ + ^x^ - \x'' + • • • = ^17 — ctn-^x.
[a;2<l.]
780. tan-ix = ^-i + -^-p^3+-.-. [a;^>l.]
2 a; 3 a;'' 5ic^ ■- -*
„oi 1 ttI 1 1-3 1.3-5
781. sec~^a; = — —
2 a; 6a;' 2.4.5a;^ 2 • 4 • 6 • 7 a;'
= ^TT — csc"~^a;. [a;^>l.]
782. log sin a; = log a; - i a;^ - ^\^ x^ - ^^l^ a;«
22"-ii?,„_ia;2»
?i
(2^0!
[a;2<7r2.]
783. log cos a; = - -^ a;2 - J^ x^ - J3. a;« - ^\\^ x»
027J-1 /92n _ -|\ 7? ^2n
?i (2 /«) ! L * J
784. log tan a; = log x + ^ a;- + /^ *"* + ^§§5 ^^
/92n-l _ -|\ 02« » „2n
w(2w)! L -i J
•yft*; Bin. 1 . ,^' Sa-" %x^ 2,x^ mx' ^
[x2 < 00.]
SERIES. 93
786. e-- = e(^l-- + — --g^+---J- [a:^<cc.]
787. e-^ = 1 + ^ + ^ + ^V ^' + ^ + • • •. [:.^<i7r^]
788. .sin-^ = l+:. + | + ^ + ^4----. [x^<l.]
789. e'^""'" = l+^ + |'-f -||-' • [^^<1.J
/yt'i /ytb /yti
*Aj tAj \Aj
790. sinha; = a; + - + gj + y^H . [a;=^<^.]
rg\^ /yt^ /yiO ™0
»A^ «Ay »A.' »Ay
791. coshx = l + - + - + - + -+---. [a;2<Q0.]
2! 4! 6! 8!
£c .^. ^. ^A ^ X
3
792. tanh x = (2' — 1) 2^^! — - (2* - 1) 2*^3 t] H
= :S[(-l)"-^22«(22»-l)^2„-ia;'"-V(2w)!].
793. ctnh ic = - (1 + 2 [(- 1)"-' 22» j52„_i cc2V(2 ti) !]).
[a;2<7r2.]
794. sech cc = 1 + 2 [(- 1)» ^2„ ic'V(2 »0 H- [^' < i t^'-]
795. csch X = - - (2 - 1)2 B,^^ + (2^ - 1)2 B,^
= i (1 + 2 2[(-l)"(2^"-^-l) ^2„_x 0^27(2 ^On)-
[a;2<7r2.]
796. sinh-^ = x-^x^ + ^^'-|^;|^ + --..[:«^<l.]
94
SERIES.
797. tanh-ia; = x + | + ^ + ^+- • •. [a;^<l.]
798. ctnh-i ^ = --^T-s + ^5+-"' [a;2> 1.]
7QQ 1.-1 .1 1 , 1-3 1-3-5
7yy. cscn x-^ 2.3.x3'^2.4 Sx^ 2.4.6.7 •x'-''^ "'•
800. J|^V-'(Zic =a; - ^ x^ + ^ - -^^ + • • •. [a;2< QO.]
801.
n5 /yt9 /j^lo
J^'COS (X') <to = 0! - g^, + g^ - j^, + • ■ • . [^» < «.]
'•Xt
802. I ^— ^ = - ^ +
+ x'' a a + b a + 2b a + 3b
+ • • •.
803. f(x + h) = f(x) + h •/' (x + Oh).
h'
804. f(x + h) = f(x) + A ./' (X) + - /" (x)
805. /(x -h A) =f(x) -{-h-f'(x) + -/"(x)
nl ^ '
+
A"
+ (;^^^-^^~^^""'--^"^'^ + ^^>
806. /(x + A, y + A;) =/(x, li) + 7i/'^(a; + ^7i, y 4 ^A;)
+ ^/'^(x + ^A, y + ^A;).
807.
SERIES. 96
i (,/2ii^ + 3 « ^£(^ + 3 M' m^
3!V ^^^ ^ ^Z/-^-^' ^-^-^y
+j,M^y...^E^
■.f(x, y) + QiD^ + ^^.)/(^, y) + I; {hD. + A-i),)y (a;, 2/)
ft •
««« . 4 r . TTCc , , . Sttcc , , . Stto; n
808. 1 = - sm h i sin 1- i sm + • • • •
[0<ic<c.]
o«« 2 (■ r . Tra; , . 2 ttsc , , . 3 ttcc ~|
809. x = — sin A^ sm h ^ sm • ■ •
TT [_ c c c J
l-c<x<c.']
c ^^-r TTiP , 1 3 7rar 1 ^irx ~|
810. ^ = ___^cos- + 3,cos— + ^cos-^ + ---J-
[0 < a- < c.-]
2'irx
«,-. . 2rr/7r2 4\ . irx tt^ .
811- -^=^LVT"l/"~~"2'^
/tt^ 4\ . 37ra; tt^ . 4
+ U-3-3>^"-^-4^^"-
+ (—--, )sin — + ••• • [0 <»•<('.]
2 TTir . 1 3 irx
5 5V c
C^ 4 C^ r TTX 1
812. a;"-^ = r cos ^ cos h ^ cos
32 6-
42 — c
_l,cosi^ + ---]- l-c<x<c.-]
m
SERIES AND PRODUCTS.
813 log sin ^x = — log 2 — cos x — ^ cos 2 a; — -^ cos 3x — • ■ '.
[0<a;<^7r.]
814. log cos ^x = — log 2 + cos x — ^ cos 2x + -^ cos 3 .r — • • • .
[0<ir<i7r.]
815. /(x) = i ^0 -I" ^1 cos — - -\- b^ cos (- • • •
. Tra; . 2 Tra- ^ ^ -,
+ «! sin h agsm 1- • • •, f— c<.x<.c.\
c c •- -"
where ^,„ = - I / (a) cos da,
1 r+'^
C •>' — c
?ft7ra ,
sm da.
816. sin^=^ 1
['
W,L
817. cos^
.27r.
1-1
2^
=[-(i')'] [-©)■] [-(.")•]
2^-4^6^ •• • (2m)^(2?>i + 2) TT
12.32.52 ... (^2 m + 1)-' 2
2^ -4' -6" • • • (2my(2m + l)
12.32.52 • • • (2m + ly
819.
^"^^~2"n!l 2(2/^ + 2)^2-
a"
(2 /i + 2) 2 • 4 (2 ?? + 2) (2 7i + 4)
.r
2 • 4 ■ 6 (2 ;i -h 2) (2 w + 4) (2 ft + 6)
TT +
}
820.
DERIVATIVES. 97
F. — Dekivatives.
d (au) a du
dx dx
R91 d{u + v) _du dv
dx dx dx
833. -^^; — - = V - — \- u--'
dx dx dx
fu\ du dv
QOQ \'V _ dx dx
dx v^
824 ^/('^) = df(u) ^ du ^
dx du dx
d\f{u) _ df (Ihi (lf_ dtl
dx^ du dx^ du? dx^
826. ^ = ?^a;«-^
dx
827. ^ = e-.
dx
oo« -^*" du .
829. '^^x^{l^\o^,x).
d{\o^^x) 1 log„e
830.
o?iC x • logg a a;
_-, c? sm a;
831. — ; = cos X.
dx
d cos X
833. — ; = — sm X.
dx
98 DERIVATIVES.
833. — % = sec^ic.
ax
834. — = — csc^ic.
ax
835. — ' = tan x • sec x.
ax
836. — ; — = — ctn x ■ CSC x.
ax
««-, (I sin~^a; 1
837. ; --
ax
(I cos~^a:;
dx
d tan~^aj
dx
d etn^^.-B
dx
d sec~-^a;
dx
d csc~'rK
838.
839.
840.
841.
842.
Vl-x^
-1
Vl-x«
1
1+x'
1
l+x2
1
x Vcc^ — 1
1
-.„ d^rahx .
843. ; = cosh a;.
dx
_. - (^ cosh a; . , "
844. — ^- — = smh x.
ax
o._ d tanh x . „
845. ; = sech^ x.
dx
846. ^li^=-csch^...
dx
DERIVATIVES. 99
847. = — seen x ■ tanh x.
ax
848. = — cscli X ■ ctnli x.
ax
_._ d sinh~^a:; 1
849. -
dx -yjx^ + 1
d cosh" ^ a; 1
850.
851.
852.
853.
854.
855. ^^£j{x)dx = f{h)
dx -^x'-l
d tanh~ ^x _ 1
dx 1 — x^
d ctnh^^a:; _ 1
d sech~^a; — 1
dx X Vl — x^
d csch~^a; — 1
dx X Va;"' + 1
d ^^
db.
856. ^fy(x)dx = -f(a).
857. jjjix, c) dx =£l)J(x,c) . dx +f(b, c) g - f(a, c) ^•
r.,ro d«(u-v) d"u , dv d"-'^u
858. — ^^ = V ■ h n-- ; r
dx" dx" dx dx"-^
n(n — V) d^v d"-^ic , d'^v
^ 2! dx^ dx^-^ dx''
859. If f(x, y, z, • • •) is a homogeneous function of the wth
order, so that /(Ax, Xy, \z, • • °) ^ X"/(x, y, z, • • •),
x-DJ+y.DJ+z.DJ+-'- = 7if.
100 DERIVATIVES.
860. Ux = <f>(y),
dy _ 1 A _ _ < ^"(y)
15
861. If sc = f{t) and ?/ = (^ {t),
dy^£({) d'y_ f'(t)-<f>"(t)-f"(t).<f>'(t)
dx f'(ty dx' [f'(t)Y
862. Uf(x,y) = 0,
dy ^ V >^f_ DJ
dx dx ' dy D^f
dhi _ D,y • ipjY - 2 D^BJ. DJ. DJ+ D,y ■ jDJ)
dx' {DJ)
863. If y =f(u, v), u = <f>{x), and v = \p{x),
d£ ^^i dM di <ii^
cZx gw dx ^ dv dx ^«/ + ^ ^"/'
^_ay /^iA" ay du dv_ dy^
dx^ du^ \dxj du ■ do dx dx d''
df d?u df dh
du dx^ dv dx^
■v \dxj
= u'' ■ D\f +2u'-v'- L„ DJ+ v'^ ■ DJ^f
+ u".I)J+v".DJ.
864. If f{x, y, z) = 0, D^--^- DJjDJ,
DJz = -lD:'f.{DJf
- 2 DJ. DJ- DJ)J^ D,y{DJy^J{DJf,
Djy^z = - ID^DJ- {Djy - DJDJ. D^DJ
+ DJ. DJ. DJ)J+ DJ. DJ. DJ^Iipjy
DERIVATIVES. 101
865. If r=</)(w, v), u=fx{x,y), and v=f^{x, u),
D^ V= d: <i> . (D,uy+ d: </, . {D^vf + 2 D^D, ^ . D,u ■ D,v
+ 2 i>„D,«/> • {_D,u ■ D^v + D,jU ■ D.y]
In the special case, u^r = Va-'^ + y'^, v = 6 = tan~^ (^//a-),
we have D^x = cos = x/ Va;'^ + ?/- ; D^y = sin = y I Va;- + ?/^;
Z>0X = — r sin = — y \ D^y = r cos ^ = a; ;
D^r = a; / Va;^ + y'^ = cos ^ ; Z>^r = ?// •va^M-l? = sin ^j
I^:P=-y I (^' + 2/') = - sin ^/r- ;
Dyd=x / (x^ + y^) = C0Bd /r\ and
Z>/ F + X*,;- F = Z>,^ F + - -i), V+\- De" V.
866. If F= «^(?<, v)' ''*=/iO^ ^). and v = f^{r, 6),
2>,2 F + ^ • A- F + i ■ A' F = i)„2 F- [(Aw)' + ^^$^~\
r r
102 DERIVATIVES.
867. If V=4>{u, V, iv), u =fi{x, ij, z), V =f^{x, y, z), and
w=fs(x, y, z),
I)JV= BJ^V. (R,icy + D^'V. (D^vf + DJV- (D^wf
B^ V + D; V + D,'V= D^'V- [{D^uy + (!>,«)'+ (A^O'l
+ i),^ r[(D,^(;)^ + (D^tvy + (D^wy^
+ 2 i)„A F- [ J>,tt . X>,^; + D,it ■ J>,v + JD^u ■ A«]
+ 2 1),A,V- IB.v ■ D^w + D,^v . D,jw + D,v ■ B.w^
+ 2 i)„.i)„ V- [Bjv ■ JDji + D,/a ■ D^u + D^^v ■ i),w]
+ D„F.[i)> + I>/« + A'^*]
In particular, if
a; = r sin 6 cos <^, y = ?' sin 6 sin ^, z = r cos ^,
30 that M = j-^^ = a;2 + y2 _^ ,-s^ i; - ^ = tan-^ ( Va;^ -f- f/z),
w~^~ tan~^ {y /x), we have
Z>,.« = cos ^ = s/ Vx^ + ?/ + .v^ ;
D^x = sin ^ cos ^ = x / VxM- y^ + ^^ ;
DERIVATIVES.
103
r^ sin 6
V.y — sin 6 sin <^ = y I V.^'- + if + z^\
I)qZ = — r sin ^ = — Vcc^ + if ;
jD^ = r eos ^ cos <^ = zx / ^j^ + y^ :
J)^y z= r cos ^ sin ^ = zi/ / Vo;- + 3/^ ;
B^z^O;
D^x = — r sin ^ sin ^ = — y ;
X)^y = r sin ^ cos <^ = x ^
7)_r = s/r = cos 6\
D£ = - Vx-2 + //r^ = - sin 0/r j
Z)^?- = X /r = sin ^ cos <^ ;
J)^0 = xz/7-^ 'Vx^ -{- f = COS cos <;?>/r;
^;«<^= -I//(^^ + f)= -sin<^/r sin^j
-^/ = y/-'" = sin ^ sin <^ ;
Z)y$ = ^1/ / i^ Va;^ + y"^ — cos ^ sin ^/r;
X>j,<^ = X / (x^ + U') = cos ^ /r sin 6 ;
(i>,r)^ + (i>,r)^ + (D^rf = 1 ;
(DAY + (A,<^)^ + (A<^)' - 1 /'-^ siii'^ ;
D^^V + I),fV + B.^V
D,{r- ■ B, V) ■sinO + ^^ + A(sin 6 ■ A F)
104
DERIVATIVES.
868. If X =fi{u, v), y =/2(m, v), z =fz{u, v),
D^ =
_ D..f,-Dj\~DJ,-D,.f,
869. If X =/(«, u), and y = ^(z, u),
Byz = BJlip,^. BJ- BJ. i>„<^).
870. If F^ (x, y, z, u, v) = 0,
F^ (x, y, z, u, v) — 0, and F^ (x, y, z, u, v) = 0,
B^
B,F, B^F, B^F,
B^F, B^F, B,F,
B,F, B^F, B,Fz
B,F, B^F, B^F,
B,F, B^F, B,F,
B,F, B^F, B,F,
871. If F^ (x, y, z) = 0, and F^ (x, y, z) = 0,
cly dz
B,F, . B,F^ - B,F, . B,F, BJ\ • B,^F, - B,F^ ■ B,,F,
dx
ByF^.B,F^-B^F^-B,F,'
If each of the quantities y^, y^, yz, • • • 2/„ is a function of
the n variables x^i x^, x^, ' • • x^, the determinant,
B^^yi B^^j^ B^^y^ • • •
B^^y^ B^^j^ B^^j^ ■ ' •
B^^y„ B^,^y„ D^ij,^ ■ • • B^j„
873.
DERIVATIVES. 105
is called the functional determinant or the Jacobian of the
^s with respect to the a;'s and is denoted by the expression,
g,jr2 g(yi> y2> 2/3, •• • Vn) . g (^^^1, ^2, X^, • ' ' X„) ^ ^
d {Xi, a-2, Xs, ' • ■ Xn) d (3/1, y2, Vz, • • • Vn) ~
d (Vl, ?/2, Vs, ■ ■ • Vn) . g (^1, ^2, ^3, • • • ^„)
g (^Ij ^2) ^3j ■ ■ ' ^n) g (^1? "^25 X^, • • • X^
^ d (]/l, 3/2, y?., ■ • • Vn)
(Xi, X^f Xg, • • • X^)
If the ?/'s are not all independent but are connected by an
equation of the form (f> (jji, ?/2, ys, ' ' ■ y„) = 0, the Jacobian
of the ?/'s with respect to the cc's vanishes identically ; and,
conversely, if the Jacobian vanishes identically, the ?/'s are
connected by one or more relations of the above-mentioned
form.
The directional derivative of any scalar point function, u,
at any point, P, in any fixed direction PQ\ is the limit, as
PQ approaches zero, of the ratio of «q — Up to PQ, where
^ is a point on the straight line PQ' between P and ^'. The
gradie7it, h^, of the function ti at P is the directional deriva-
tive of M at P taken in the direction in which w increases
most rapidly. This direction is normal to the surface of
constant m which passes through P.
874. K' = {D,u)' + {D^n)' + {D^uf.
The directional derivative of any scalar point function at
any point in any given direction is evidently equal to the
product of the gradient and the cosine of the angle between
the given direction and that in which the function increases
most rapidly.
106 MISCELLANEOUS FORMULAS.
The normal derivative, at any point, P, of a point function
u, taken with respect to another point function v, is the limit
as P(^ approaches zero of the ratio of «q — tip to Vq — Vp,
where ^ is a point so chosen on the normal at P of the
surface of constant v which passes through P, that Vq — Vp
is positive. If (u, v) denotes the angle between the directions
in which u and v increase most rapidly, the normal derivatives
of u with respect to v, and of v with respect to u may be
written
h^^ cos (?<, v) -7- 7ij,, and A„ • cos (w, v) -v- ^„
respectively. If A„ = h^, these derivatives are equal.
Gr. — Miscellaneous Formulas.
If s is a plane analytic closed curve, n its normal drawn
from within outwards, and dA the element of plane area
within s, the usual integral transformation formulas for the
functions u and v which, with their derivatives of the first
order, are continuous everywhere within s, may be written —
875. I M • cos (x, n) ds = \ i D^u ■ dA.
876. j [w • cos (x, 7i) + V ■ cos (?/, w)] ds=^ C C(D^ti + DyV) dA.
Sn. Jb„u .ds= C C (B/u + Dyhi) dA.
878. j'^iD.n . Djj + D,^u ■ D^v) dA
= Ju ■ D^v -ds- C Cu (Z)> + D,fv) dA
= Cv . D^u .ds-^ifv {B^u + D,fv) dA.
879. f C\ {D^u ■ Djo + D,;ii ■ IJ,/-) dA = Cxu- D„v ■ ds
-ff ' U'. (^ • A'O + ^. (^ • ^>M dA
MISCELLANEOUS FORMULAS. 107
If ^ and 7} are two analytic functions which define a set of
orthogonal curvilinear coordinates, and if (^, n) and {-q, n)
represent the angles between n and the directions in which
^ and 7], respectively, increase most rapidly.
880. ^j'h^ ■ \ • A ( r ) ^^ =X" ■ ^°^ ^'^' ^^^ ^^'
881. ^ ^ h^ h^-DA^jdA =fu . cos (i, n) ds.
882. If r is the distance from a fixed point, Q, in the coordi-
nate plane,
/cos (v 71) cl'S
— '-^^^ — —— = 0, TT, or 2 TT, according as Q is without,
on, or within s.
If a9 is an analytic closed surface, n its normal drawn from
within outwards, and dr the element of volume shut in by S,
the usual integral transformation formulas may be written —
883. r Cu cos (x, 71) dS= C C C D^u ■ dr.
884. I I [y« cos {x, n) + v cos (y, n) + tv cos (z, n)"] dS
= f f fi^x'if' + I>y^ + D,w)dT.
885. r rz)„« • (7s = ( ( r (^/« + ^2,''* + a'«) <^t.
886. j" ^ j" (7),^« . D^v + i>^7f . D^v + i),-a . X>,y) dr
= r r« -D^v-dS- C C Cu (Bj'v + Z>/y + n^^) dr
= f f^- ^nU dS- C C C r {Dju + Dfa + D.hi) dr.
108 MISCELLANEOUS FOKMULAS.
887. fff>^ {D^u ■ D,v + DyU ■ D^v + D,u ■ D,v) dr
- ^ ^ ^ vlB^iXB^u) + D^iXD^u) + D,{XD,u)^dr,
If I, rj, I are three analytic functions which define a system
of orthogonal curvilinear coordinates,
889. jyj"/'| • ^ • h^ ■ Dr, (j;^) ^^ =ff"' ■ cos (^7, ?0 ^'S^-
890. ////^f • hr, .\-D^ {irrh) '^^ =//" • ^°' (^' '') '^'^-
891. If r is the distance from a fixed point, Q,
/cos ^?' ??'^
-j—^ dS = 0, 2 TT, or 4 TT according as Q is without,
on, or within S.
Stokes's Theorem, — The line integral, taken around a
closed curve, of the tangential component of a vector point
function, is equal to the surface integral, taken over a surface
bounded by the curve, of the normal component of the curl of
the vector, the direction of integration around the curve form-
ing a right-handed screw rotation about the normals.
If X, Y, Z are the components of the vector,
892. C{Xdx + Ydy + Zdz) = C C[(D,,Z - I), Y) cos (x, n)
+ (I),X ~ D,Z) cos (t/, n)
+ (B, Y - DyX) cos {z, n)-] dS.
MISCELLANEOUS FORMULAS. 109
Equations 893 to 897 give Poisson's Equation in orthogonal
Cartesian, in cylindrical, in spherical, and in orthogonal curvi
linear coordinates.
893. v2r=Z»/r+X>/F+ A'^=-4 7rp.
1
894. ~I),(r.D,V) + ^-De'r+l),^V=-4.7rp.
895. sme.DJr^-D,.V) + ^^
^ ^ sm 6
+ Dg (sin ei)0V) = - A Trpr'' sin 6.
896. 7i/ ■D^^V+ /i,2 ■D^W+ hi ■ Dl V
397. ;,,./,,. /.,{i),(,^^^.A^')+ A
_rL
•i>„F
y.,A, ^
H. — Certain Constants.
7r = 3.14159 26535 89793
logio7r = 0.49714 98726 94134
- = 0.31830 98861 83791
TT
TT^ = 9.86960 44010 89359
V^ = 1.77245 38509 05516
logio 2 = 0.30102 99956 63981
e = 2.71828 18284 59045
logio e = 0.43429 44819 03252
log, 10 = 2.30258 50929 94046
log^2 = 0.69314 71805 59945
togiologio e = 9.63778 43113 00537
log^7r = 1.14472 98858 49400
+ ^dj±-^<r)^ = -*^P
110 FORMULAS OF INTEGEATION.
I. — General Fokmulas of Integration.
F and / represent functions of x, and F\ /', F'\ f", their
first and second derivatives with respect to x.
898
899
900
901,
Cf' •/• dx = Ff- Cf-/' ■ dx.
. C{Fy-F'-dx = (Fy+^/(u + i).
C(aF + by ■ F' ■ dx = (aF + by + ^/a (n + 1\
J(F +fy -dx ^Jf(f +fy-'dx +Jf{F +f)-'dx.
902. CF'/(Fy-dx = -l/(n-l)(Fy-\ Cf'/F- dx = logF.
903. /(^'-Z- F.f ')/(/)''. dx = F/f.
. f{F'-f- F-f)IFf. dx = log {FIf).
J dx _ _1^ r dx d_ r dx
F-{x^ -a?) ~2^,J F-{x-a) ~ 2~aJ F ■ {x + a)
r dx _ r dx r dx
■JF(F±f)~' J F.f^J f{F±f)'
/pi fly,
, = (2^aF+b)/a.
y/aF + b
C F'-dx , ,„ /-— ,
I , = log {F + -^I^' + a).
/ Fdx _ a r dx b f dx
{F+a)(F + b) ~ a-bJ Y+~a. ~ a-bJ F+b
r F-dx _ r dx r fdx
J (F+fY~ J (F+ f)"-' ^ J TF
910
{F+fy j(F+fy-^ ^{F+fy
911 r_Zji^ = l -I'lZ r F'-dx ^ 1 ^ qF-p
FOEMULAS OF INTEGllATION. Ill
913. f f! '"". =- tan-',
J F^ -\- a} a \ a
F'-dx 1. ^lF\
aF — b
aF +b
914. I -—- — T7, = TTT ^°S
rF^^dx _ r F^-dx r t^"-.
or./;
{F'^ -h) V 2(i'^»+^'
F'^d^__ 1
+ hF~ h '"^ aF +b
917. " .. . , „ = T loR-
919. f , ^' = y sec- ' ( ^
r
J — INTEGRALS Useful in the Theory of Alternating
Currents.
922
923
924
I sill ((nt -{- (f>)dt = • cos (wt + </)).
. I cos (oit -\- ^)dt = -- sin (oit -\- <(>).
/I 1
sm^(u)t + <f)dt^-t — -— sin 2 (i^t + </>).
112 AUXLLIAIIY FORMULAS.
925. / sin (uyf + <^) ■ cos (wf + cl>)dt = — - sm'^(wt + <}>).
/I 1
cos2(o)^ -\- <t>)dt = -t + -— sin 2 (u)t + cfi).
927. fsin (.ot -\- X) ■ sin (a>^ + /x) dt = ^^^ — ^ (o)^)
sin (<i)t + A) • cos (wt + /^)
2 (1)
_„_ I . , , , ^ , sin (<Dt + A) • sin (wt -\- a)
928. / sin (oyt + A^ • cos (oyt 4- m) dt = ^^ — ^^-^ ^^ — ^^-^
/ sin ((at + A) • cos (wt 4- /u.) f/i^
sin (/Lt — A)
(-0-
929. / cos (wt + A) • cos (wt + /x) fZi = ^"^ ^^ ^ (w^)
sin (<«>^ + A) • cos (wt + A)
2 w
^„^ C . , . , ^ , sin r???f — n?' + A — Atl
930. / sm (»^ ^ -I- A) • sm (ni + yu) </^ = ^-—-^ =4 ^
J 2 (?« — n)
sin [w/ + ?t^ 4- A + /"■]
2 (m + w)
931. f cos (mt + A) • cos int + ix) dt = — ~- ; — ;
J ' 2 (/?» + n)
sm\^nit — nt -{- X — fi']
2 (in — ri)
o,«« r ■ , s , s , cos r??2i + nt + X+ ul}
932. / sin r^/^i + A) • cos ^nr + n) dt = ^-— 9= , ^ ^-'
2 (m + ?'')
cos [?wi — nt -\- X — fJi]
2 (»i - n)
J. I sin (?//i -}- A) • cos (ni + fx) dt =
AUXILIARY FOIlxMULAS.
113
933
'. I cos ((tit + A + vi.r) ■ COS (oj^ + A — III J') dx
= cos'^(w^ + A)
inx -\- sin m.r • cos iiix
mx — sin mx • cos inx
2«i.
-sin2(w;' + A)
m- sin(w/'4-<^)4-?i- cos(w)'+<^) = V»r+H.2- sin(a)f+<^+c)^
\ where tan c = n/m.
III ■ sin(<Df+^)— w • cos(w!'+<^)= V//r+?i-^ • sm{(t)t-\-<i>—d).
934.
/
e^-^^"^'(h^
-h^ cl
W + c^
^e
.(-b±ci)t
-ht
= 77 ; [(c ■ sin ct — b- cos ct) =F «' G> ■ sin r?' 4- c • cos cf)!
0^ 4-c^^
r.-ht
—= [sin (cf - 8) =F '■ • cos (c;^ - 8)]
935. j e"' ■ cos (<of + tf>) (If
a^ + o,
where tan S = /^/c.
^ [to sin ((of + ^) 4- « • cos ((of + <^)]
= — -==:^ cos [to;' + </) — tan ^(a)/«)].
936. C (>"'■ sin ((of + <f>) (It
^ai
a^+(o
2 [a • sin (wf + ^) — o> • cos ((of + <^ )]
937. /^[e''' • sin ((of + <i>)'fdt
■ sin \_(of + <^ — tan ^(w/a)].
4
1 to • sin 2 ( o)/' 4- <^) + « ■ cos 2 (ojj'
'^ + <^) ]
a cf'^ + ft)"^ J
1 cosr2a)?' + 2(^-tan-i(ai/Q:)] "
(X
Va"^ +
U)
'1
114
AUXILIARY FORMULAS.
938. fie"' ■ COS (wt + <i»)ydt
„2at
1 o) • sin 2 (u)^ + ^) + (T • cos 2 (wt + ^) "
a
«
«-^ +
1 cos[2W + 2<^-taii-Xco/«)] "|
V^7+^
In the case of a direct trigonometric function of (wt + <!>),
T = 2 tt/o) is called the ^>erio(^ or the c//f^e. The mean
value for any whole number of periods, reckoned from any
epoch, of sin (wt + <^), cos (wt + (j>), or sin (cof + ^) • cos (wf + </>),
is zero, whereas the mean value for any whole number of half
periods, reckoned from any epoch, of either sin'-^ (wt + <f)) or
cos^((ot + (f>) is one half. The mean value of sin (w.') from
^ = to ;* = ^ T, or of cos (<ot) from - ^ T to + i T, is 2/7r
or 0.6366.
The mean value, for any number of whole periods, of either
sin(a)?'+X) • sin(co^+/x) or cos(cu?'+X) • <-os(wf+fi) is h ■ cos(A— /x),
while the mean value of sin(wi; + A) ■ cos ((Dt -\- /x) is ^ sin (\ — fi).
TABLES. 115
INTERPOLATION.
If values of an analytic f unction, /(x), are given in a table for a number
of values of the argument x, separated from one another consecutively by
the constant small interval, 5, the differences between successive tabular
values of the function are called ^rsi tabular differences, the differences of
these first differences, second tabular differences, and so on. The tabular
differences of the first, second, third, and fourth orders corresponding to
z= a are
Ai=/(a + 5)-/(a),
A2 =/(a + 2 5) - 2 -/(a + 5) +/(a),
A3 =f{a + 3 5) - 3 -/{a + 2 5) + 3 -/{a + 0) -f{a),
A4 =/(a + 4 5) - 4 .f{a + 3 5) + 6 -/(a + 2 5) - 4 ./(a + 5) +/(a),
where / (a) is any tabulated value.
The value of the function for x = {a + h), v?here h = kS, is
Z.X ^. V , * k{k-l) ^ k(k-l)(k-2) ^
f{a + h) =/(a) + A; • Ai + ^^^ • A2 + ^ ^ '- ■ A3
_^ Mfc-l)(fc-2)(A:-3) _^_.^^^^
116
TABLES.
"he Probability Integral.
1
( 2
'0
dx.
)
X
1
2
3
4
5
6
7
8
9
0.00
0.00000
00113
00226
00339
00451
00564
00677
00790
00903
01016
0.01
0.01128
01241
01354
01467
01580
01792
01805
01918
02031
02144
0.02
0.02256
02369
02482
02595
02708
02820
02933
03046
03159
03271
0.03
0.03384
03497
03610
03722
03835
03948
04060
04173
04286
04398
0.04
0.04511
04624
04736
04849
04962
05074
05187
05299
05412
05525
0.05
0.05637
05750
05862
05975
06087
06200
06312
06425
06537
06650
0.06
0.06762
06875
06987
07099
07212
07324
07437
07549
07661
07773
0.07
0.07SS6
07998
08110
08223
08335
08447
08559
08671
08784
08896
0.08
0.09008
09120
09232
09344
09456
09568
09680
09792
09904
10016
0.09
0.10128
10240
10352
10464
10576
10687
10799
10911
11023
11135
0.10
0.11246
11358
11470
11581
11693
11S05
11916
12028
12139
12251
0.11
0.12362
12474
12585
12697
12808
12919
13031
13142
13253
13365
0.12
0.13476
135S7
13698
13809
13921
14032
14143
14254
14365
14476
0.13
0.14587
14698
14809
14919
15030
15141
15252
15363
15473
15584
0.14
0.15695
15805
15916
16027
16137
16248
16358
16468
16579
16689
0.15
0.16800
16910
17020
17130
17241
17351
17461
17571
17681
17791
0.16
0.17901
18011
18121
18231
18341
18451
18560
18670
18780
18890
0.17
0.18999
19109
19218
19328
19437
19547
19656
19766
19875
19984
0.18
0.20094
20203
20312
20421
20530
20639
20748
20857
20966
21075
0.19
0.21184
21293
21402
21510
21619
21728
21836
21945
22053
22162
0.20
0.22270
22379
22487
22595
22704
22812
22920
23028
23136
23244
0.21
0.23352
23460
23568
23676
23784
23891
23999
24107
24214
24322
0.22
0.24430
24537
24645
24752
24859
24967
25074
25181
25288
25395
0.23
0.25502
25609
25716
25823
25930
26037
26144
26250
26357
26463
0.24
0.26570
26677
26783
26889
26996
27102
27208
27314
27421
27527
0.25
0.27633
27739
27845
27950
28056
28162
28268
28373
28479
28584
0.26
0.28690
28795
28901
29006
29111
29217
29322
29427
29532
29637
0.27
0.29742
29847
29952
30056
30161
30266
30370
30475
30579
30684
0.28
0.30788
30892
30997
31101
31205
31309
31413
31517
31621
31725
0.29
0.31828
31922
32036
32139
32243
32346
32450
32553
32656
32760
0.30
0.32863
32966
33069
33172
33275
33378
33480
33583
33686
33788
0.31
0.33891
33993
34096
34198
34300
34403
34505
34607
34709
34811
0.32
0.34913
35014
35116
35218
35319
35421
35523
35624
35725
35827
0.33
0.35928
36029
36130
36231
36332
36433
36534
36635
36735
36836
0.34
0.36936
37037
37137
37238
37338
37438
37538
37638
37738
37838
0.35
0.37938
38038
38138
38237
38337
38436
38536
38635
38735
38834
0.36
0.38933
39032
39131
39230
39329
39428
39526
39625
39724
39822
0.37
0.39921
40019
40117
40215
40314
40412
40510
40608
40705
40803
0.38
0.40901
40999
41096
41194
41291
41388
41486
41583
41680
41777
0.39
0.41874
41971
42068
42164
42261
42358
42454
42550
42647
42743
0.40
0.42839
42935
43031
43127
43223
43319
43415
43510
43606
43701
0.41
0.43797
43892
43988
44083
44178
44273
44368
44463
44557
44652
0.42
0.44747
44841
44936
45030
45124
45219
45313
45407
45501
45595
0.43
0.45689
45782
45876
45970
46063
46157
46250
46343
46436
46529
0.44
0.46623
46715
46808
46901
46994
47086
47179
47271
47364
47456
0.45
0.47548
47640
47732
47824
47916
48008
48100
48191
48283
48374
0.46
0.48466
48557
48648
48739
48830
48921
49012
49103
49193
49284
0.47
0.49375
49465
49555
49646
49736
49826
49916
50006
50096
50185
0.48
0.50275
50365
50454
50543
50633
50722
50811
50900
50989
51078
0.49
0.51167
51256
51344
51433
51521
51609
51698
51786
51874
51962
1
TABLES. 117
The Probability Integral.
123456789
0.52050 52138 52226 52313 52401 52488 52576 52663 52750 52837
0.52924 53011 53098 53185 53272 53358 53445 53531 53617 5370+
0.53790 53876 53962 54048 54134 54219 54305 54390 54476 54561
0.54646 54732 54817 54902 54987 55071 55156 55241 55325 55410
0.55494 55578 55662 55746 55830 55914 55998 56082 56165 56249
0.56332 56416 56499 56582 56665 56748 56831 56914 56996 57079
0.57162 57244 57326 57409 57491 57573 57655 57737 57818 57900
0.57982 58063 58144 58226 58307 58388 58469 58550 58631 58712
0.58792 58873 58953 59034 59114 59194 59274 59354 59434 59514
0.59594 59673 59753 59832 59912 59991 60070 60149 60228 60307
0.60386 60464 60543 60621 60700 60778 60856 60934 61012 61090
0.61168 61246 61323 61401 61478 61556 61633 61710 61787 61864
0.61941 62018 62095 62171 62248 62324 62400 62477 62553 62629
0.62705 62780 62856 62932 63007 63083 63158 63233 63309 63384
0.63459 63533 63608 63683 63757 63832 63906 63981 64055 64129
0.64203 64277 64351 64424 64498 64572 64645 64718 64791 64865
0.64938 65011 65083 65156 65229 65301 65374 65446 65519 65591
0.65663 65735 65807 65878 65950 66022 66093 66165 66236 66307
0.66378 66449 66520 66591 66662 66732 66803 66873 66944 67014
0.67084 67154 67224 67294 67364 67433 67503 67572 67642 67711
0.67780 67849 67918 67987 68056 68125 68193 68262 68330 68398
0.68467 68535 68603 68671 68/38 68806 68874 68941 69009 69076
0.69143 69210 69278 69344 69411 69478 69545 69611 6%78 69744
0.69S10 69877 69943 70009 70075 70140 70206 70272 70337 70403
0.70468 70533 70598 70663 70728 70793 70858 70922 70987 71051
0.71116 71180 71244 71308 71372 71436 71500 71563 71627 71690
0.71754 71817 71880 71943 72006 72069 72132 72195 72257 72320
0.72382 72444 72507 72569 72631 72693 72755 72816 72878 72940
0.73001 73062 73124 73185 73246 73307 73368 73429 73489 73550
0.73610 73671 73731 73791 73851 73911 73971 74031 74091 74151
0.74210 74270 74329 74388 74447 74506 74565 74624 74683 74742
0.74800 74859 74917 74976 75034 75092 75150 75208 75266 75323
0.75381 75439 75496 75553 75611 75668 75725 75782 75839 75896
0.75952 76009 76066 76122 76178 76234 76291 76347 76403 76459
0.76514 76570 76626 76681 76736 76792 76847 76902 76957 77012
0.77067 77122 77176 77231 77285 77340 77394 77448 77502 77556
0.77610 77664 77718 77771 77825 77878 77932 77985 78038 78091
0.78144 78197 78250 78302 78355 78408 78460 78512 78565 78617
0.78669 78721 78773 78824 78876 78928 78979 79031 79082 79133
0.79184 79235 79286 79337 79388 79439 79489 79540 79590 79641
0.79691 79741 79791 79841 79891 79941 79990 80040 80090 80139
0.80188 80238 80287 80336 80385 80434 80482 80531 80580 80628
0.80677 80725 80773 80822 80870 80918 80966 81013 81061 81109
0.81156 81204 81251 81299 81346 81393 81440 81487 81534 81580
0.81627 81674 81720 81767 81813 81859 81905 81951 81997 82043
0.82089 82135 82180 82226 82271 82317 82362 82407 82452 82497
0.82542 82587 82632 82677 82721 82766 82810 82855 82899 82943
0.82987 83031 83075 83119 83162 83206 83250 83293 83337 83380
0.83423 83466 83509 83552 83595 83638 83681 83723 83766 83808
0.83851 83893 83935 83977 84020 84061 84103 84145 84187 84229
118
TABLES.
The Probability Integral.
"^ dx.
X
1
2
3
4
5
6
^
1
8
9
1.00
0.84270
84312
84353
84394
84435
84477
84518
84559
84600
84640
1.01
0.84681
84722
84762
84803
84843
84883
84924
84964
85004
85044
1.02
0.85084
85124
85163
85203
85243
85282
85322
85361
85400
85439
1.03
0.85478
85517
85556
85595
85634
85673
85711
85750
85788
85827
1.04
0.85865
85903
85941
85979
86017
86055
86093
86131
86169
86206
1.05
0.86244
86281
86318
86356
86393
86430
86467
86504
86541
86578
1.06
0.86614
86651
86688
86724
86760
86797
86833
86869
86905
86941
1.07
0.86977
87013
87049
87085
87120
87156
87191
87227
87262
87297
1.08
0.87333
87368
87403
87438
87473
87507
87542
87577
87611
87646
1.09
0.87680
87715
87749
87783
87817
87851
87885
87919
87953
87987
1.10
0.88021
88054
880SS
88121
88155
SS188
88221
88254
88287
88320
1.11
0.88353
88386
88419
88452
88484
88517
88549
88582
88614
88647
1.12
0.88679
88711
88743
88775
88807
8SS39
88871
88902
88934
88966
1.13
0.88997
89029
89060
89091
89122
89] 54
89185
89216
89247
89277
1.14
0.89308
89339
89370
89400
89431
89461
89492
89522
89552
89582
1.15
0.89612
89642
89672
89702
89732
89762
89792
89821
89851
89880
1.16
0.89910
89939
89968
89997
90027
90056
90085
90114
90142
90171
1.17
0.90200
90229
90257
90286
90314
90343
90371
90399
90428
90456
1.18
0.90484
90512
90540
90568
90595
90623
90651
90678
90706
90733
1.19
0.90761
90788
90815
90843
90870
90897
90924
90951
90978
91005
1.20
0.91031
91058
91085
91111
91138
91164
91191
91217
91243
91269
1.21
0.91296
91322
91348
91374
91399
91425
91451
91477
91502
91528
1.22
0.91553
91579
91604
91630
91655
91680
91705
91730
91755
91780
1.23
0.91805
91830
91855
91879
91904
91929
91953
91978
92002
92026
1.24
0.92051
92075
92099
92123
92147
92171
92195
92219
92243
92266
1.25
0.92290
92314
92337
92361
92384
92408
92431
92454
92477
92500
1.26
0.92524
92547
92570
92593
92615
92638
92661
92684
92706
92729
1.27
0.92751
92774
92796
92819
92841
92863
92885
92907
92929
92951
1.28
0.92973
92995
93017
93039
93061
93082
93104
93126
93147
93168
1.29
0.93190
93211
93232
93254
93275
93296
93317
93338
93359
93380
1.30
0.93401
93422
93442
93463
93484
93504
93525
93545
93566
93586
1.31
0.93606
93627
93647
93667
93687
93707
93727
93747
93767
93787
1.32
0.93807
93826
93846
93866
93885
93905
93924
93944
93963
93982
1.33
0.94002
94021
94040
94059
94078
94097
94116
94135
94154
94173
1.34
0.94191
94210
94229
94247
94266
94284
94303
94321
94340
94358
1.35
0.94376
94394
94413
94431
94449
94467
94485
94503
94521
94538
1.36
0.94556
94574
94592
94609
94627
94644
94662
94679
94697
94714
1.37
0.94731
94748
94766
94783
94800
94817
94834
94851
94868
94885
1.38
0.94902
94918
94935
94952
94968
94985
95002
95018
95035
95051
1.39
0.95067
95084
95100
95116
95132
95148
95165
95181
95197
95213
1.40
0.95229
95244
95260
95276
95292
95307
95323
95339
95354
95370
1.41
0.95385
95401
95416
95431
95447
95462
95477
95492
95507
95523
1.42
0.95538
95553
95568
95582
95597
95612
95627
95642
95656
95671
1.43
0.95686
95700
95715
95729
95744
95758
95773
95787
95801
95815
1.44
0.95830
95844
95858
95872
95886
95900
95914
95928
95942
95956
1.45
0.95970
95983
95997
96011
96024
96038
96051
96065
96078
96092
1.46
0.96105
96119
96132
96145
96159
96172
96185
96198
96211
96224
1.47
0.96237
96250
96263
96276
96289
96302
96315
96327
96340
96353
1.48
0.96365
96378
96391
96403
96416
96428
96440
96453
96465
96478
1.49
0.96490
96502
96514
96526
96539
96551
96563
96575
96587
96599
TABLES.
119
The Probability Integral.
X
2 4 6 8
X
2 4 6 8
1.50
0.96611 96634 96658 96681 96705
2.00
0.99532 99536 99540 99544 99548
1.51
0.96728 96751 96774 96796 96819
2.01
0.99552 99556 99560 99564 99568
1.52
0.96841 96S64 96886 96908 96930
2.02
0.99572 99576 99580 99583 99587
1.53
0.96952 96973 96995 97016 97037
2.03
0.99591 99594 99598 99601 99605
1.54
0.97059 97080 97100 97121 97142
2.04
0.99609 99612 99616 99619 99622
1.55
0.97162 97183 97203 97223 97243
2.05
0.99626 99629 99633 99636 99639
1.56
0.97263 97283 97302 97322 97341
2.06
0.99642 99646 99649 99652 99655
1.57
0.97360 97379 9739S 97417 97436
2.07
0.99658 99661 99664 99667 99670
1.58
0.97455 97473 97492 97510 97528
2. OS
0.99673 99676 99679 996S2 99685
1.59
0.97546 97564 97582 97600 97617
2.09
0.99688 99691 99694 99697 99699
1.60
0.97635 97652 97670 97687 97704
2.10
0.99702 99705 99707 99710 99713
1.61
0.97721 97738 97754 97771 97787
2.11
0.99715 99718 99721 99723 99726
1.62
0.97804 97820 97836 97852 97868
2.12
0.99728 99731 99733 99736 99738 '
1.63
0.978S4 97900 97916 97931 97947
2.13
0.99741 99743 99745 99748 99750
1.64
0.97962 97977 97993 98008 98023
2.14
0.99753 99755 99757 99759 99762
1.65
0.9S03S 98052 98067 98082 98096
2.15
0.99764 99766 99768 99770 99773
1.66
0.98110 98125 98139 98153 98167
2.16
0.99775 99777 99779 997S1 99783
1.67
0.98181 98195 98209 98222 98236
2.17
0.99785 99787 99789 99791 99793
1.68
0.98249 9S263 9S276 98289 98302
2.18
0.99795 99797 99799 99801 99803
1.69
0.9S315 98328 9S341 98354 98366
2.19
0.99805 99806 99808 99810 99812
1.70
0.98379 98392 98404 98416 98429
2.20
0.99814 99815 99817 99819 99821
1.71
0.98441 9S453 98465 98477 98489
2.21
0.99822 99824 99826 99827 99829
1.72
0.98500 98512 98524 98535 98546
2.22
0.99831 99832 99834 99836 99S37
1.73
0.98558 98569 98580 98591 98602
2.23
0.99839 99840 99842 99843 99845
1.74
0.98613 98624 98635 98646 98657
2.24
0.99846 99848 99849 99851 99852
1.75
0.98667 98678 98688 98699 98709
2.25
0.99854 99855 9985 7 99858 99859
1.76
0.98719 9S729 98739 98749 98759
2.26
0.99861 99862 99863 99865 99866
1.77
0.98769 98779 98789 98798 98808
2.27
0.99867 99869 99870 99871 99873
1.78
0.98817 98827 9SS36 98846 98855
2.28
0.99874 99875 99876 99877 99879
1.79
0.98864 98873 98882 98891 98900
2.29
0.99880 99881 99882 99883 99885
1.80
0.98909 98918 98927 98935 98944
2.30
0.99886 99887 99888 99889 99890
1.81
0.98952 98961 98969 98978 98986
2.31
0.99891 99892 99893 99894 99896
1.82
0.98994 99003 99011 99019 99027
2.32
0.99897 99898 99899 99900 99901
1.83
0.99035 99043 99050 99058 99066
2.33
0.99902 99903 99904 99905 99906
1.84
0.99074 99081 99089 99096 99104
2.34
0.99906 99907 99908 99909 99910
1.85
0.99111 99118 99126 99133 99140
2.35
0.99911 99912 99913 99914 99915
1.86
0.99147 99154 99161 99168 99175
2.36
0.99915 99916 99917 99918 99919
1.87
0.99182 99189 99196 99202 99209
2.37
0.99920 99920 99921 99922 99923
l.SS
0.99216 99222 99229 99235 99242
2.38
0.99924 99924 99925 99926 99927
1.89
0.99248 99254 99261 99267 99273
2.39
0.99928 99928 99929 99930 99930
1.90
0.99279 99285 99291 99297 99303
2.40
0.99931 99932 99933 99933 99934
1.91
0.99309 99315 99321 99326 99332
2.41
0.99935 99935 99936 99937 99937
1.92
0.99338 99343 99349 99355 99360
2.42
0.99938 99939 99939 99940 99940
1.93
0.99366 99371 99376 99382 99387
2.43
0.99941 99942 99942 99943 99943
1.94
0.99392 99397 99403 99408 99413
2.44
0.99944 99945 99945 99946 99946
1.95
0.99418 99423 99428 99433 99438
2.45
0.99947 99947 99948 99949 99949
1.96
0.99443 99447 99452 99457 99462
2.46
0.99950 99950 99951 99951 99952
1.97
0.99466 99471 99476 99480 99485
2.47
0.99952 99953 99953 99954 99954
1.98
0.99489 99494 99498 99502 99507
2.48
0.99955 99955 99956 99956 99957
1.99
0.99511 99515 99520 99524 99528
2.49
0.99957 99958 99958 99958 99959
2.00
0.99532 99536 99540 99544 99548
2.50
0.99959 99960 99960 99961 99961
120
TABLES.
The Probability Integral.
2
-x2
)
X
1
2
3
4
5
6
7
8
9
2.5
0.99959
99961
99963
99965
99967
99969
99971
99972
99974
99975
2.6
0.99976
99978
99979
99980
99981
99982
99983
99984
99985
99986
2.7
0.99987
99987
99988
99989
99989
99990
99991
99991
99992
99992
2.8
0.99992
99993
99993
99994
99994
99994
99995
99995
99995
99996
2.9
0.99996
99996
99996
99997
99997
99997
99997
99997
99997
99998
3.0
0.99998
99998
99998
99998
99998
99998
99998
99998
99999
99999
The value, /, of the Probability Integral may always be found from the convergent series
, 1 / afi a;5 a-'
/= — = [x — ,
■v^V 3-l!^5-2! 7-3!
■)■
but for large values of x, the semiconvergent series
x-^-a\ 2x2 (2x2)3 (2x2)3^ )
is convenient.
i
TABLES.
121
Values of the Complete Elliptic Integrals, JTand E, for Different
Values of the Modulus, k.
K= p. ^^ ; E= pVl-A;2sin-z.
»/o Vl — kr sin'' z *yo
dz.
sin-ifc
E
E
sin-iA;
K
E
1
sin-ifc
K
E
0°
1.5708
1.5708
30°
1.6858
1.4675
60°
2.1565
1.2111
1°
1.5709
1.5707
31°
1.6941
1.4608
61°
2.1842
1.2015
2°
1.5713
1.5703
32°
1.7028
1.4539
62°
2.2132
1.1920
3°
1.5719
1.5697
33°
1.7119
1.4469
63°
2.2435
1.1826
4°
1.5727
1.5689
34°
1.7214
1.4397
64°
2.2754
1.1732
5°
1.5738
1.5678
35°
1.7312
1,4323
65°
2.308S
1.1638
6°
1.5751
1.5665
36°
1.7415
1.4248
66°
2.3439
1.1545
7°
1.5767
1.5649
37°
1.7522
1.4171
67°
2.3809
1.1453
8°
1.5785
1.5632
38°
1.7633
1.4092
68°
2.4198
1.1362
9°
1.5S05
1.5611
39°
1.7748
1.4013
69°
2.4610
1.1272
10°
1.5828
1.5589
40°
1.7868
1.3931
70°
2.5046
1.1184
11°
1.5854
1.5564
41°
1.7992
1.3849
71°
2.5507
1.1096
12°
1.5882
1.5537
42°
1.8122
1.3765
72°
2.5998
1.1011
13°
1.5913
1.5507
43°
1.8256
1.3680
73°
2.6521
1.0927
14°
1.5946
1.5476
44°
1.8396
1.3594
74°
2.7081
1.0S44
15°
1.5981
1.5442
45°
1.S541
1.3506
75°
2.7681
1.0764
16°
1.6020
1.5405
46°
1.S691
1.3418
76°
2.8327
1.0686
17°
1.6061
1.5367
47°
1.SS48
1.3329
77°
2.9026
1.0611
18°
1.6105
1.5326
48°
1.9011
1.3238
78°
2.9786
1.0538
19°
1.6151
1.5283
49°
1.9180
1.3147
79°
3.0617
1.0468
20°
1.6200
1.5238
50°
1.9356
1.3055
80°
3.1534
1.0401
21°
1.6252
1.5191
51°
1.9539
1.2963
81°
3.2553
1.0338
22°
1.6307
1.5141
52°
1.9729
1.2S70
82°
3.3699
1.0278
23°
1.6365
1.5090
53°
1.9927
1.2776
83°
3.5004
1.0223
24°
1.6426
1.5037
54°
2.0133
1.26S1
84°
3.6519
1.0172
25°
1.6490
1.4981
55°
2.0347
1.2587
85°
3.8317
1.0127
26°
1.6557
1.4924
56°
2.0571
1.2492
86°
4.0528
1.0086
27°
1.6627
1.4864
57°
2.080+
1.2397
87°
4.3387
1.0053
28°
1.6701
1.4803
58°
2.101-7
1.2301
88°
4.7427
1.0026
29°
1.6777
1.4740
59°
2.1300
1.2206
89°
5.4349
1.0008
122
TABLES.
Values of F{k, <t>) for Certain Values of k and ^t,
dz
^<*'*>=XV
A;2 sin2 z
<t>
a = sin-^k.
0°
1(P
15°
30°
45°
60°
75°
80°
90°
1°
0.0174
0.0174
0.0174
0.0174
0.0174
0.0174
0.0174
0.0174
0.0174
2°
0.0349
0.0349
0.0349
0.0349
0.0349
0.0349
0.0349
0.0349
0.0349
3°
0.0524
0.0524
0.0524
0.0524
0.0524
0.0524
0.0524
0.0524
0.0524
40
0.0698
0.0698
0.069S
0.0698
0.0698
0.0699
0.0699
0.0699
0.0699
5°
0.0873
0.0873
0.0873
0.0873
0.0873
0.0S74
0.0874
0.0874
0.0874
10°
0.1745
0.1746
0.1746
0.1748
0.1750
0.1752
0.1754
0.1754
0.1754
15°
0.2618
0.2619
0.2620
0.2625
0.2633
0.2641
0.2646
0.2647
0.2648
20°
0.3491
0.3493
0.3495
0.3508
0.3526
0.3545
0.3559
0.3562
0.3564
25°
0.4363
0.4367
0.4372
0.4397
0.4433
0.4470
0.4498
0.4504
0.4509
30°
0.5236
0.5243
0.5251
0.5294
0.5356
0.5422
0.5474
0.5484
0.5493
35°
0.6109
0.6119
0.6132
0.6200
0.6300
0.6408
0.6495
0.6513
0.6528
40°
0.6981
0.6997
0.7016
0.7116
0.7267
0.7436
0.7574
0.7604
0.7629
45°
0.7854
0.7876
0.7902
0.8044
0.8260
0.8512
0.8727
0.8774
0.8814
50°
0.8727
0.8756
0.8792
0.8982
0.9283
0.9646
0.9971
1.0044
1.0107
55°
0.9599
0.9637
0.9683
0.9933
1.0337
1.0848
1.1331
1.1444
1.1542
60°
l.(H72
1.0519
1.0577
1.0896
1.1424
1.2125
1.2837
1.3014
1.3170
65°
1.1345
1.1402
1.1474
1.1869
1.2545
1.3489
1.4532
1.4810
1.5064
70°
1.2217
1.2286
1.2373
1.2853
1.3697
1.4944
1.6468
1.6918
1.7354
75°
1.3090
1.3171
1.3273
1.3846
1.4S79
1.6492
1.8714
1.9468
2.0276
80°
1.3963
1.4056
1.4175
1.4846
1.6085
1.8125
2.1339
2.2653
2.4362
85°
1.4835
1.4942
1.5078
1.5850
1.7308
1.9826
2.4366
2.6694
3.1313
86°
1.5010
1.5120
1.5259
1.6052
1.7554
2.0172
2.5013
2.7612
3.3547
87°
1.5184
1.5297
1.5439
1.6253
1.7801
2.0519
2.5670
2.8561
3.6425
88°
1.5359
1.5474
1.5620
1.6454
1.8047
2.0867
2.6336
2.9537
4.0481
89°
1.5533
1.5651
1.5S01
1.6656
1.8294
2.1216
2.7007
3.0530
4.7414
90°
1.5708
1.5828
1.5981
1.6858
1.8541
2.1565
2.7681
3.1534
Inf.
TABLES.
123
Values of E(k, 4>) for Certain Values of k and 0.
E{k, ^) = I Vl - fc2 sin2 z • dz.
:
a = sin-i&.
0°
10°
15°
30°
45°
60°
75°
80°
90°
P
0.0174 0.0174
0.0174
0.0174
0.0174
0.0174
0.0174
0.0174
0.0174
2»
0.0349 0.0349
0.0349
0.0349
0.0349
0.0349
0.0349
0.0349
0.0349
3»
0.0524 0.0524
0.0524
0.0524
0.0524
0.0523
0.0523
0.0523
0.0523
40
0.0698
0.0698
0.0698
0.0698
0.0698
0.0698
0.0698
0.0698
0.0698
5°
0.0873
0.0873
0.0873
0.0872
0.0872
0.0872
0.0S72
0.0872
0.0872
10°
0.1745
0.1745
0.1745
0.1743
0.1741
0.1739
0.1737
0.1737
0.1736
15°
0.2618
0.2617
0.2616
0.2611
0.2603
0.2596
0.2590
0.2589
0.2588
20°
0.3491
0.3489
0.3486
0.3473
0.3456
0.3438
0.3425
0.3422
0.3420
25°
0.4363
0.4359
0.4354
0.4330
0.4296
0.4261
0.4236
0.4230
0.4226
30°
0.5236
0.5229
0.5221
0.5179
0.5120
0.5061
0.5016
0.5007
0.5000
35°
0.6109
0.6098
0.6085
0.6019
0.5928
0.5833
0.5762
0.5748
0.5736
40°
0.6981
0.6966
0.6947
0.6851
0.6715
0.6575
0.6468
0.6446
0.6428
45°
0.7854
0.7832
0.7806
0.7672
0.7482
0.7282
0.7129
0.7097
0.7071
50°
0.8727
0.8698
0.8663
0.8483
0.8226
0.7954
0.7741
0.7697
0.7660
55°
0.9599
0.9562
0.9517
0.9284
0.8949
0.8588
0.8302
0.8242
0.8192
60°
1.0472
1.0426
1.0368
1.0076
0.9650
0.9184
0.8808
0.8728
0.8660
65°
1.1345
1.12SS
1.1218
1.0858
1.0329
0.9743
0.9258
0.9152
0.9063
70°
1.2217
1.2149
1.2065
1.1632
1.0990
1.0266
0.9652
0.9514
0.9397
75°
1.3090
1.3010
1.2911
1.2399
1.1635
1.0759
0.9992
0.9814
0.9659
80°
1.3963
1.3S70
1.3755
1.3161
1.2266
1.1225
1.0282
1.0054
0.9848
85°
1.4835
1.4729
1.4598
1.3919
1.2889
1.1673
1.0534
1.0244
0.9962
86°
1.5010
1.4901
1.4767
1.4070
1.3012
1.1761
1.0581
1.0277
0.9976
87°
1.5184
1.5073
1.4936
1.4221
1.3136
1.1848
1.0628
1.0309
0.9986
88°
1.5359
1.5245
1.5104
1.4372
1.3260
1.1936
1.0674
1.0340
0.9994
89°
1.5533
1.5417
1.5273
1.4524
1.3383
1.2023
1.0719
1.0371
0.9998
90«
1.5708
1.5589
1.5442
1.4675
1.3506
1.2111
1.0764
1.0401
1
1.0000
124
TABLES.
Hyperbolic Functions.
1,
e^
e-^
sinhx
coshx
gdx
0.00
1.0000
1.0000
0.0000
1.0000
o!oooo
.01
1.0100
0.9900
.0100
1.0000
0.5729
.02
1.0202
.9802
.0200
1.0002
1.1458
.03
1.0305
.9704
.0300
1.0004
1.7186
.04
1.0408
.9608
.0400
1.0008
2.2912
.05
1.0513
.9512
.0500
1.0013
2.8636
.06
1.0618
.9418
.0600
1.0018
3.4357
.07
1.0725
.9324
.0701
1.0025
4.0074
.08
1.0833
.9231
.0801
1.0032
4.5788
.09
1.0942
.9139
.0901
1.0041
5.1497
.10
1.1052
.9048
.1002
1.0050
5.720
.11
1.1163
.8958
.1102
1.0061
6.290
.12
1.1275
.8869
.1203
1.0072
6.859
.13
1.1388
.8781
.1304
1.0085
7.428
.14
1.1503
.8694
.1405
1.0098
7.995
.15
1.1618
.8607
.1506
1.0113
8.562
.16
1.1735
.8521
.1607
1.0128
9.128
.17
1.1853
.8437
.1708
1.0145
9.694
.18
1.1972
.8353
.1810
1.0162
10.258
.19
1.2092
.8270
.1911
1.0181
10.821
.20
1.2214
.8187
.2013
1.0201
11.384
.21
1.2337
.8106
.2115
1.0221
11.945
.22
1.2461
.8025
.2218
1.0243
12.505
.23
1.2586
.7945
.2320
1.0266
13.063
.24
1.2712
.7866
.2423
1.0289
13.621
.25
1.2840
.7788
.2526
1.0314
14.177
.26
1.2969
.7711
.2629
1.0340
14.732
.27
1.3100
.7634
.2733
1.0367
15.285
.28
1.3231
.7558
.2837
1.0395
15.837
.29
1.3364
.7483
.2941
1.0423
16.388
.30
1.3499
.7408
.3045
1.0453
16.937
.31
1.3634
.7334
.3150
1.0484
17.484
.32
1.3771
.7261
.3255
1.0516
18.030
.33
1.3910
.7189
.3360
1.0549
18.573
.34
1.4049
.7118
.3466
1.0584
19.116
.35
1.4191
.7047
.3572
1.0619
19.656
.36
1.4333
.6977
.3678
1.0655
20.195
.37
1.4477
.6907
.3785
1.0692
20.732
.38
1.4623
.6839
.3892
1.0731
21.267
.39
1.4770
.6771
.4000
1.0770
21.800
.40
1.4918
.6703
.4108
1.0811
22.331
.41
1.5068
.6636
.4216
1.0852
22.859
.42
1.5220
.6570
.4325
1.0895
23.386
.43
1.5373
.6505
.4434
1.0939
23.911
.44
1.5527
.6440
.4543
1.0984
24.434
.45
1.5683
.6376
.4653
1.1030
24.955
.46
1.5841
.6313
.4764
1.1077
25.473
.47
1.6000
.6250
.4875
1.1125
25.989
.48
1.6161
.6188
.4986
1.1174
26.503
.49
1.6323
.6126
.5098
1.1225
27.015
0.50
1.648f
0.6065
0.5211
1.1276
27?524
Note. —This table is talien from Prof. Byerly's Treatise on Fourier's Series, published by Messrs.
Oiim& Co.
TABLES.
125
Hyperbolic Functions.
X
e-^
er-x
sinhx
coshx
gdx
0.50
1.6487
0.6065
0.5211
1.1276
27!524
.51
1.6653
.6005
.5324
1.1329
28.031
.52
1.6820
.5945
.5438
1.1383
28.535
.53
1.6989
.5886
.5552
1.1438
29.037
.54
1.7160
.5827
.5666
1.1494
29.537
.55
1.7333
.5770
.5782
1.1551
30.034
.56
1.7507
.5712
.5897
1.1609
30.529
.57
1.7683
.5655
.6014
1.1669
31.021
.58
1.7860
.5599
.6131
1.1730
31.511
.59
1.8040
.5543
.6248
1.1792
31.998
.60
1.8221
.5488
6367
1.1855
32.483
.61
1.8404
.5433
.6485
1.1919
32.965
.62
1.8589
.5379
.6605
1.1984
33.444
.63
1.S776
.5326
.6725
1.2051
33.921
.64
1.8965
.5273
.6846
1.2119
34.395
.65
1.9155
.5220
.6967
1.2188
34.867
.66
1.9348
.5169
.7090
1.2258
35.336
.67
1.9542
.5117
.7213
1.2330
35.802
-68
1.9739
.5066
.7336
1.2402
36.265
-69
1.9937
.5016
.7461
1.2476
36.726
.70
2.0138
.4966
.7586
1.2552
37.183
-71
2.0340
.4916
.7712
1.2628
37.638
-72
2.0544
.4867
.7838
1.2706
38.091
-73
2.0751
.4819
.7966
1.2785
38.540
.74
2.0959
.4771
.8094
1.2865
38.987
.75
2.1170
.4724
.8223
1.2947
39.431
.76
2.1383
.4677
.8353
1.3030
39.872
-77
2.1598
.4630
.8484
1.3114
40.310
.78
2.1815
.4584
.8615
1.3199
40.746
-79
2.2034
.4538
.8748
1.3286
41.179
-80
2.2255
.4493
.8881
1.3374
41.608
-81
2.2479
.4449
.9015
1.3464
42.035
.82
2.2705
.4404
.9150
1.3555
42.460
-83
2.2933
.4360
.9286
1.3647
42.881
-84
2.3164
.4317
.9423
1.3740
43.299
-85
2.3396
.4274
.9561
1.3835
43.715
.86
2.3632
.4232
.9700
1.3932
44.128
.87
2.3869
.4190
.9840
1.4029
44.537
.88
2.4109
.4148
.9981
1.4128
44.944
.89
2.4351
.4107
1.0122
1.4229
45.348
.90
2.4596
.4066
1.0265
1.4331
45.750
.91
2.4843
.4025
1.0409
1.4434
46.148
-92
2.5093
.3985
1.0554
1.4539
46.544
-93
2.5345
.3946
1.0700
1.4645
46.936
-94
2.5600
.3906
1.0847
1.4753
47.326
-95
2.5857
.3867
1.0995
1.4862
47.713
-96
2.6117
.3829
1.1144
1.4973
48.097
.97
2.6379
.3791
1.1294
1.5085
48.478
.98
2.6645
.3753
1.1446
1.5199
48.857
.99
2.6912
.3716
1.1598
1.5314
49.232
1.00
2.7183
0.3679
1.1752
1.5431
49!60S
siiih X = tan gd x ; cosh a: = sec gd a; ; tanh x = sin gd x.
126
TABLES.
Hyperbolic Functions.
X
I .si nil X
I cosh X
X
isinhx
I cosh X
X
isinhx
I cosh X
1.00
0.0701
0.1884
1.50
0.3282
0.3715
2.00
0.5595
0.5754
1.01
.0758
.1917
1.51
.3330
.3754
2.01
.5640
.5796
1.02
.0815
.1950
1.52
.3378
.3794
2.02
.5685
.5838
1.03
.0871
.1984
1.53
.3426
.3833
2.03
.5730
.5880
1.04
.0927
.2018
1.54
.3474
.3873
2.04
.5775
.5922
1.05
.0982
.2051
1.55
.3521
.3913
2.05
.5820
.5964
1.06
.1038
.2086
1.56
.3569
.3952
2.06
.5865
.6006
1.07
.1093
.2120
1.57
.3616
.3992
2.07
.5910
.6048
1.08
.1148
.2154
1.58
.3663
.4032
2.08
.5955
.6090
1.09
.1203
.2189
1.59
.3711
.4072
2.09
.6000
.6132
1.10
.1257
.2223
1.60
.3758
.4112
2.10
.6044
.6175
1.11
.1311
.2258
1.61
.3805
.4152
2.11
.6089
.6217
1.12
.1365
.2293
1.62
.3852
.4192
2.12
.6134
.6259
1.13
.1419
.2328
1.63
.3899
.4232
2.13
.6178
.6301
1.14
.1472
.2364
1.64
.3946
.4273
2.14
.6223
.6343
1.15
.1525
.2399
1.65
.3992
.4313
2.15
.6268
.6386
1.16
.1578
.2435
1.66
.4039
.4353
2.16
.6312
.6428
1.17
.1631
.2470
1.67
.4086
.4394
2.17
.6357
.6470
1.18
.1684
.2506
1.68
.4132
.4434
2.18
.6401
.6512
1.19
.1736
.2542
1.69
.4179
.4475
2.19
.6446
.6555
1.20
.1788
.2578
1.70
.4225
.4515
2.20
.6491
.6597
1.21
.1840
.2615
1.71
.4272
.4556
2.21
.6535
.6640
1.22
.1892
.2651
1.72
.4318
.4597
2.22
.6580
.6682
1.23
.1944
.2688
1.73
.4364
.4637
2.23
.6624
.6724
1.24
.1995
.2724
1.74
.4411
.4678
2.24
.6668
.6767
1.25
.2046
.2761
1.75
.4457
.4719
2.25
.6713
.6809
1.26
.2098
.2798
1.76
.4503
.4760
2.26
.6757
.6852
1.27
.2148
.2835
1.77
.4549
.4801
2.27
.6802
.6894
1.28
.2199
.2872
1.78
.4595
.4842
2.28
.6846
.6937
1.29
.2250
.2909
1.79
.4641
.4883
2.29
.6890
.6979
1.30
.2300
.2947
1.80
.4687
.4924
2.30
.6935
.7022
1.31
.2351
.2984
LSI
.4733
.4965
2.31
.6979
.7064
1.32
.2401
.3022
1.82
.4778
.5006
2.32
.7023
.7107
1.33
.2451
.3059
1.83
.4824
.5048
2.33
.7067
.7150
1.34
.2501
.3097
1.84
.4870
.5089
2.34
.7112
.7192
1.35
.2551
.3135
1.85
.4915
.5130
2.35
.7156
.7235
1.36
.2600
.3173
1.86
.4961
.5172
2.36
.7200
.7278
1.37
.2650
.3211
1.87
.5007
.5213
2.37
.7244
.7320
1.38
.2699
.3249
1.88
.5052
.5254
2.38
.7289
.7363
1.39
.2748
.3288
1.89
.5098
.5296
2.38
.7333
.7406
1.40
.2797
.3326
1.90
.5143
.5337
2.40
.7377
.7448
1.41
.2846
.3365
1.91
.5188
.5379
2.41
.7421
.7491
1.42
.2895
.3403
1.92
.5234
.5421
2.42
.7465
.7534
1.43
.2944
.3442
1.93
.5279
.5462
2.43
.7509
.7577
1.44
.2993
-3481
1.94
.5324
.5504
2.44
.7553
.7619
1.45
.3041
.3520
1.95
.5370
.5545
2.45
.7597
.7662
1.46
.3090
.3559
1.96
.5415
.5687
2.46
.7642
.7705
1.47
.3138
.3598
1.97
.5460
.5629
2.47
.7686
.7748
1.48
.3186
.3637
1.98
.5505
.5671
2.48
.7730
.7791
1.49
.3234
.3676
1.99
.5550
.5713
2.49
.7774
.7833
1.50
0.3282
0.3715
2.00
0.5595
0.5754
2.50
0.7818
0.7876
TABLES.
127
Hyperbolic Functions.
X
I sinh X
I cosh X
X
isinhx
I cosh X
X
I sinh X
I cosh X
2.50
0.7S1S
0.7876
2.1 S
0.8915
0.8951
3.0
1.0008
1.0029
2.51
.7862
.7919
2.76
.8959
.8994
3.1
1.0444
1.0462
2.52
.7906
.7962
2.77
.9003
.9037
3.2
1.0880
1.0894
2.53
.7950
.8005
2.78
.9046
.9080
3.3
1.1316
1.1327
2.54
.7994
.8048
2.79
.9090
.9123
3.4
1.1751
1.1761
2.55
.8038
.8091
2.80
.9134
.9166
3.5
1.2186
1.2194
2.56
.8082
.8134
2.81
.9178
.9209
3.6
1.2621
1.2628
2.57
.8126
.8176
2.82
.9221
.9252
3.7
1.3056
1.3061
2.58
.8169
.8219
2.83
.9265
.9295
3.8
1.3491
1.3495
2.59
.8213
.8262
2.84
.9309
.9338
3.9
1.3925
1.3929
2.60
.8257
.8305
2.85
.9353
.9382
4.0
1.4360
1.4363
2.61
.8301
.8348
2.86
.9396
.9425
4.1
1.4795
1.4797
2.62
.8345
.8391
2.87
.9440
.9468
4.2
1.5229
1.5231
2.63
.8389
.8434
2.88
.9484
.9511
4.3
1.5664
1.5665
2.64
.8433
.8477
2.89
.9527
.9554
4.4
1.6098
1.6099
2.65
.8477
.8520
2 90
.9571
.9597
4.5
1.6532
1.6533
2.66
.8521
.8563
2.91
.9615
.9641
4.6
1.6967
1.6968
2.67
.8564
.8606
2.92
.9658
.9684
4.7
1.740]
1.7402
2.68
.8608
.8649
2.93
.9702
.9727
4.8
1.7836
1.7836
2.69
.8652
.8692
2.94
.9746
.9770
4.9
1.8270
1.8270
2.70
.8696
.8735
2.95
.9789
.9813
5.0
1.8704
1.8705
2.71
.8740
.8778
2.96
.9833
.9856
6.0
2.3047
2.3047
2.72
.8784
.8821
2.97
.9877
.9900
7.0
2.7390
2.7390
2.73
.8827
.8864
2.98
.9920
.9943
8.0
3.1733
3.1733
2.74
.8871
.8907
2.99
.9964
.9986
9.0
3.6076
3.6076
2.75
0.8915
0.8951
3.00
1.0008
1.0029
10.0
4.0419
4.0419
For values of x greater than 7.0, we may write, to five places of deci-
mals at least,
logio sinh X = logio cosh x = log i e^ = x (0.4342945) + 1.6989700.
The Values of e-x^ for Certain Values of x.
X
e-^
X
e-^
X
Q-X
X
Q-X
1/10
0.90484
8/10
0.44933
18/10
0.16530
5
0.00674
1/8
0.88250
9/10
0.40657
2
0.13534
11/2
0.00409
1/6
0.84648
1
0.36788
9/4
0.10540
6
0.00248
2/10
0.81873
11/10
0.33287
5/2
0.08209
13/2
0.00150
1/4
0.77880
9/8
0.32465
8/3
0.06948
7
0.00091
3/10
0.74082
12/10
0.30119
3
0.04979
15/2
0.00055
1/3
0.71653
5/4
0.28650
25/8
0.04394
8
0.00034
4/10
0.67032
13/10
0.27253
16/5
0.04076
9
0.00012
5/10
0.60653
4/3
0.26360
18/5
0.02732
10
0.00004
6/10
0.54881
14/10
0.24660
4
0.01832
11
0.00002
2/3
0.51342
3/2
0.22313
25/6
0.01550
12
0.00001
7/10
0.49659
16/10
0.20190
9/2
0.01111
13
0.00000
128
TABLES.
The Common Logarithms of e^ and e-«.
«
logioe^
logioe-^
0.00001
0.0000043429
1.9999956571
0.00002
0.0000086859
1.9999913141
0.00003
0.0000130288
1.9999869712
0.00004
0.0000173718
1.9999826282
0.00005
0.0000217147
1.9999782853
0.00006
0.0000260577
1.9999739423
0.00007
0.0000304006
1.9999695994 •
0.00008
0.0000347436
1.9999652564
0.00009
0.0000390865
1.9999609135
0.00010
0.0000434294
1.9999565706
0.00020
0.0000868589
1.9999131411
0.00030
0.0001302883
1.9998697117
0.00040
0.0001737178
1.9998262822
0.00050
0.0002171472
1.9997828528
0.00060
0.0002605767
1.9997394233
0.00070
0.0003040061
1.9996959939
0.00080
0.0003474356
1.9996525644
0.00090
0.0003908650
1.9996091350
0.00100
0.0004342945
1.9995657055
0.00200
0.0008685890
1.9991314110
0.00300
0.0013028834
1.9986971166
0.00400
0.0017371779
1.9982628221
0.00500
0.0021714724
1.9978285276
0.00600
0.0026057669
1.9973942331
0.00700
0.0030400614
1.9969599386
0.00800
0.0034743559
1.9965256441
0.00900
0.0039086503
1.9960913497
0.01000
0.0043429448
1.9956570552
0.02000
0.0086858896
1.9913141 ICH-
0.03000
0.0130288345
1.9869711655
0.04000
0.0173717793
T.9826282207
0.05000
0.0217147241
1.9782852759
0.06000
0.0260576689
1.9739423311
0.07000
0.0304006137
1.9695993863
TABLES.
129
X
logio e*
logio e-"
0.08000
0.0347435586
1.9652564414
0.09000
0.0390865034
1.9609134966
0.10000
0.0434294482
1.9565705518
0.20000
0.0868588964
1.9131411036
0.30000
0.1302883446
1.8697116554
0.40000
0.1737177928
1.8262822072
0.50000
02171472410
1.7828527590
0.60000
0.2605766891
1.7394233109
0.70000
0.3040061373
1.6959938627
0.80000
0.3474355855
1.6525644145
090000
0.3908650337
1.6091349663
1.00000
0.4342944819
1.5657055181
2.00000
0.8685889638
1.1314110362
3.00000
1.3028834457
2.6971165543
4.00000
1.7371779276
2.2628220724
5.00000
2.1714724095
3.8285275905
6.00000
2.6057668914
3.3942331086
7.00000
3.0400613733
4.9599386267
8.00000
3.4743558552
4.5256441448
9.00000
3.9086503371
4.0913496629
10.00000
4.3429448190
5.6570551810
20.00000
8.6858896381
9.3141103619
30.00000
13.0288344571
14.9711655429 ^
40.00000
17.3717792761
18.6282207239
50.00000
21.7147240952
22.2852759048
60.00000
26.0576689142
27.9423310858
70.00000
30.4006137332
31.5993862668
80.00000
34.7435585523
35.2564414477
90.00000
39.0865033713
40.9134966287
100.00000
43.4294481903
44.5705518097
200.00000
86.8588963807
87.1411036193
300.00000
130.2883445710
131.7116554290
400.00000
173.7177927613
174.2822072387
500.00000
217.1472409516
218.8527590tS4
Note •- log e^ + V = log e* + log &>. Thus, log giis-iirs - 49.139465 ISa
130
TABLES.
Five-Place Natural Logarithms.
No.
1
2
3
4
5
6
7
8
9
D.
1.00
0.0 0000
0100
0200
0300
0399
0499
0598
0698
0797
0896
100-99
1.01
0.0 0995
1094
1193
1292
1390
1489
1587
1686
1784
1882
99-98
1.02
0.0 1980
2078
2176
2274
2372
2469
2567
2664
2762
2859
98-97
1.03
0.0 2956
3053
3150
3247
3343
3440
3537
3633
3730
3826
97-96
1.04
0.0 3922
4018
4114
4210
4306
4402
4497
4593
4688
4784
96-95
1.05
0.0 4879
4974
5069
5164
5259
5354
5449
5543
5638
5733
95-94
1.06
0.0 5827
5921
6015
6110
6204
6297
6391
6485
6579
6672
94
1.07
0.0 6766
6859
6953
7046
7139
7232
7325
7418
7511
7603
93
1.08
0.0 7696
7789
7881
7973
8066
8158
8250
8342
8434
8526
93-92
1.09
0.0 8618
8709
8801
8893
8984
9075
9167
9258
9349
9430
92-91
1.10
0.0 9531
9622
9713
9803
9894
9985
*0075
0165
0256
0346
91-90
1.11
0.1 0436
0526
0616
0706
0796
0885
0975
1065
1154
1244
90-89
1.12
0.1 1333
1422
1511
1600
1689
1778
1867
1956
2045
2133
89
1.13
0.1 2222
2310
2399
2487
2575
2663
2751
2839
2927
3015
88
1.14
0.1 3103
3191
3278
3366
3453
3540
3628
3715
3802
3889
88-87
1.15
0.1 3976
4063
4150
4237
4323
4410
4497
4583
4669
4756
87-86
1.16
0.1 4842
4928
5014
5100
5186
5272
5358
5444
5529
5615
86
1.17
0.1 5700
5786
5871
5956
6042
6127
6212
6297
6382
6467
85
1.18
0.16551
6636
6721
6805
6890
6974
7059
7143
7227
7311
85-84
1.19
0.1 7395
7479
7563
7647
7731
7815
7898
7982
8065
8149
84-83
1.20
0.1 8232
8315
8399
8482
8565
8648
8731
8814
8897
9979
83
1.21
0.1 9062
9145
9227
9310
9392
9474
9557
9639
9721
9803
83-82
1.22
0.1 9885
9967 *0049
0131
0212
0294
0376
0457
0539
0620
82-81
1.23
0.2 0701
0783
0864
0945
1026
1107
1188
1269
1350
1430
81
1.24
0.21511
1592
1672
1753
1833
1914
1994
2074
2154
2234
81-80
1.25
0.2 2314
2394
2474
2554
2634
2714
2793
2873
2952
3032
80-79
1.26
0.2 3111
3191
3270
3349
3428
3507
3586
3665
3744
3823
79
1.27
0.2 3902
3980
4059
4138
4216
4295
4373
4451
4530
4608
79-78
1.28
0.2 4686
4764
4842
4920
4998
5076
5154
5231
5309
5387
78
1.29
0.2 5464
5542
5619
5697
5774
5811
5928
6005
6082
6159
77
1.30
0.2 6236
6313
6390
6467
6544
6620
6697
6773
6850
6926
77-76
1.31
0.2 7003
7079
7155
7231
7308
7384
7460
7536
7612
7687
76
1.32
0.2 7763
7839
7915
7990
8066
8141
8217
8292
8367
8443
76-75
1.33
0.2 8518
8593
8668
8743
8818
8893
8968
9043
9118
9192
75
1.34
0.2 9267
9342
9416
9491
9565
9639
9714
9788
9862
9936
75-74
1.35
0.3 0010
0085
0158
0232
0306
0380
0454
0528
0601
0675
74
1.36
0.3 0748
0822
0895
0969
1042
1115
1189
1262
1335
1408
74-73
1.37
0.3 1481
1554
1627
1700
1773
1845
1918
1991
2063
2136
73-72
1.38
0.3 2208
2281
2353
2426
2498
2570
2642
2714
2786
2858
72
1.39
0.3 2930
3002
3074
3146
3218
3289
3361
3433
3504
3576
72-71
1.40
0.3 3647
3719
3790
3861
3933
4004
4075
4146
4217
4288
71
1.41
0.3 4359
4430
4501
4572
4642
4713
4784
4854
4925
4995
71-70
1.42
0.3 5066
5136
5206
5277
5347
5417
5487
5557
5677
5697
70
1.43
0.3 5767
5837
5907
5977
6047
6116
6186
6256
6335
6395
70-69
1.44
0.3 6464
6534
6603
6672
6742
6811
6880
6949
7018
7087
69
1.45
0.3 7156
7225
7294
7363
7432
7501
7569
7638
7707
7775
69
1.46
0.3 7844
7912
7981
8049
8117
8186
8254
8322
8390
8458
68
1.47
0.3 8526
8594
8662
8730
8798
8866
8934
9001
9069
9137
68
1.48
0.3 9204
9272
9339
9407
9474
9541
9609
9676
9743
9810
68-67
1.49
0.3 9878
9945
*0012
0079
0146
0213
0279
0346
0413
0480
67
1.50
0.4 0547
0613
0680
0746
0813
0879
0946
1012
1078
1145
67-€6
1
2
3
4
5
6
7
8
9
TABLES.
131
Five-Place Natural Logarithms.
No.
1
2
3
4
5
7
8
9
D.
1.50
0.4 0547
0613
0680
0746
0813
0879
0946
1012
1078
1145
67-66
1.51
0.41211
1277
1343
1409
1476
1542
1608
1673
1739
1805
66
1.52
0.4 1871
1937
2003
2068
2134
2199
2265
2331
2396
2461
66-65
1.53
0.4 2527
2592
2657
2723
2788
2853
2918
2983
3048
3113
65
1.54
0.4 3178
3243
3308
3373
3438
3502
3567
3632
3696
3761
65-64
1.55
0.4 3825
3890
3954
4019
4083
4148
4212
4276
4340
4404
64
1.56
0.4 4469
4533
4597
4661
4725
4789
4852
4916
4980
5044
64
1.57
0.4 5108
5171
5235
5298
5362
5426
5489
5552
5616
5679
64-63
1.58
0.4 5742
5S06
5869
5932
5995
6058
6122
6185
6248
6310
63
1.59
0.4 6373
6436
6499
6562
6625
6687
6750
6813
6875
6938
63
1.60
0.4 7000
7063
7125
7188
7250
7312
7375
7437
7499
7561
62
1.61
0.4 7623
7686
7748
7810
7872
7933
7995
8057
8119
8181
63
1.62
0.4 8243
8304
8366
8428
8489
8551
8612
8674
8735
8797
62-61
1.63
0.4 8858
8919
8981
9042
9103
9164
9225
9287
9348
9409
61
1.64
0.4 9470
9531
9592
9652
9713
9774
9835
9896
9956 *0017
61
1.65
0.5 0078
013S
0199
0259
0320
0380
0441
0501
0561
0622
61-60
1.66
■ 0.5 0682
0742
0802
0862
0922
0983
1043
1103
1163
1222
60
1.67
0.5 1282
1342
1402
1462
1522
1581
1641
1701
1760
1820
60
1.68
0.5 1879
1939
1998
2058
2117
2177
2236
2295
2354
2414
60-59
1.69
0.5 2473
2532
2591
2650
2709
2768
2827
2886
2945
3004
59
1.70
0.5 3063
3122
3180
3239
3298
3357
3415
3474
3532
3591
59-58
1.71
0.5 3649
3708
3766
3825
3883
3941
4000
4058
4116
4174
58
1.72
0.5 4232
4291
4349
4407
4465
4523
4581
4639
4696
4754
58
1.73
0.5 4812
4870
4928
4985
5043
5101
5158
5216
5274
5331
58-57
1.74
0.5 5389
5446
5503
5561
5618
5675
5/oj
5790
5847
5904
57
1.75
0.5 5962
6019
6076
6133
6190
6247
6304
6361
6418
6475
57
1.76
0.5 6531
6588
6645
6702
6758
6815
6872
6928
6985
7041
57
1.77
0.5 7098
7154
7211
7267
7324
7380
7436
7493
7549
7605
56
1.78
0.5 7661
7718
7774
7830
7886
7942
7998
8054
8110
8166
56
1.79
0.5 8222
8277
8333
8389
8445
8501
8556
8612
8667
8723
56
1.80
0.5 8779
8834
8890
8945
9001
9056
9111
9167
9222
9277
56-55
1.81
0.5 9333
9388
9443
9498
9553
9609
9664
9719
9774
9829
55
1.82
0.5 9884
9939
9993
*004S
0103
0158
0213
0268
0322
0377
55
1.83
0.6 0432
0486
0541
0595
0650
0704
0759
0813
0868
0922
55-54
1.84
0.6 0977
1031
1085
1139
1194
1248
1302
1356
1410
1464
54
1.85
0.61519
1573
1627
1681
1735
1788
1842
1896
1950
2004
54
1.86
0.6 2058
2111
2165
2219
2272
2326
2380
2433
2487
2540
54-53
1.87
0.6 2594
2647
2701
2754
2808
2861
2914
2967
3021
3074
53
1.88
0.6 3127
3180
3234
3287
3340
3393
3446
3499
3552
3605
53
1.89
0.6 3658
3711
3763
3816
3869
3922
3975
4027
4080
4133
53
1.90
0.6 4185
4238
4291
4343
4396
4448
4501
4553
4606
4658
53-52
1.91
0.6 4710
4763
4815
4867
4920
4972
5024
5076
5128
5180
52
1.92
0.6 5233
5285
5337
5389
5441
5493
5545
5596
5648
5700
52
1.93
0.6 5752
5804
5856
5907
5959
6011
6062
6114
6166
6217
52
1.94
0.6 6269
6320
6372
6423
6475
6526
6578
6629
6680
6732
52-51
1.95
0.6 6783
6834
6885
6937
6988
7039
7090
7141
7192
7243
51
1.96
0.6 7294
7345
7396
7447
7498
7549
7600
7651
7702
7753
51
1.97
0.6 7803
7854
7905
7956
8006
8057
8107
8158
8209
8259
51
1.98
0.6 8310
8360
8411
8461
8512
8562
8612
8663
8713
8763
50
1.99
0.6 8813
8864
8914
8964
9014
9064
9115
9165
9215
9265
50
2.00
0.6 9315
9365
9415
9465
9515
9564
9614
9664
9714
9764
50
1
2
3
4
5
6
7
8
9
132
TABLES.
Five-Place Natural Logarithms.
No.
1
2
3
4
5
6
7
8
9
D.
2.00
0.6 9315
9365
9415
9465
9515
9564
9614
9664
9714
9764
50
2.01
0.6 9813
9863
9913
9963 -
*^0012
0062
0112
0161
0211
0260
50
2.02
0.7 0310
0359
0409
0458
0508
0557
0606
0656
0705
0754
49
2.03
0.7 0804
0853
0902
0951
1000
1050
1099
1148
1197
1246
49
2.04
0.7 1295
1344
1393
1442
1491
1540
1589
1638
1686
1735
49
2.05
0.7 1784
1833
1881
1930
1979
2028
2076
2125
2173
2222
49
2.06
0.7 2271
2319
2368
2416
2465
2513
2561
2610
2658
2707
49-48
2.07
0.7 2755
2803
2851
2900
2948
2996
3044
3092
3141
3189
48
2.08
0.7 3237
3285
3333
3381
3429
3477
3525
3573
3621
3669
48
2.09
0.7 3716
3764
3812
3860
3908
3955
4003
4051
4098
4146
48
2.10
0.7 4194
4241
4289
4336
4384
4432
4479
4527
4574
4621
48-47
2.11
0.7 4669
4716
4764
4811
4858
4905
4953
5000
5047
5094
47
2.12
0.7 5142
5189
5236
5283
5330
5377
5424
5471
5518
5565
47
2.13
0.7 5612
5659
5706
5753
5800
5847
5893
5940
5987
6034
47
2.14
0.7 6081
6127
6174
6221
6267
6314
6361
6407
6454
6500
47
2.LS
0.7 6547
6593
6640
6686
6733
6779
6825
6872
6918
6965
47-46
2.16
0.7 7011
7057
7103
7150
7196
7242
7288
7334
7381
7427
46
2.17
0.7 7473
7519
7565
7611
7657
7703
7749
7795
7841
7887
46
2.18
0.7 7932
7978
8024
8070
8116
8162
8207
8253
8299
8344
46
2.19
0.7 8390
8436
8481
8527
8573
8618
8664
8709
8755
8800
46-45
2.20
0.7 8846
8891
8937
8982
9027
9073
9118
9163
9209
9254
45
2.21
0.7 9299
9344
9390
9435
9480
9525
9570
9615
9661
9706
45
2.22
0.7 9751
9796
9841
9886
9931
9976 *0021
0066
0110
0155
45
2.23
0.8 0200
0245
0290
0335
0379
0424
0469
0514
0558
0603
45
2.24
0.8 0648
0692
0737
0781
0826
0871
0915
0960
1004
1049
45-44
2.25
0.8 1093
1137
1182
1226
1271
1315
1359
1404
1448
1492
44
2.26
0.8 1536
1581
1625
1669
1713
1757
1802
1846
1890
1934
44
2.27
0.8 1978
2022
2066
2110
2154
2198
2242
2286
2330
2374
44
2.28
0.8 2418
2461
2505
2549
2593
2637
2680
2724
2768
2812
44
2.29
0.8 2855
2899
2942
2986
3030
3073
3117
3160
3204
3247
44-43
2.30
0.8 3291
3334
3378
3421
3465
3508
3551
3595
3638
3681
43
2.31
0.8 3725
3768
3811
3855
3898
3941
3984
4027
4070
4114
43
2.32
0.8 4157
4200
4243
4286
4329
4372
4415
4458
4501
4544
43
2.33
0.8 4587
4630
4673
4715
4758
4801
4844
4887
4930
4972
43
2.34
0.8 5015
5058
5101
5143
5186
5229
5271
5314
5356
5399
43
2.35
0.8 5442
5484
5527
5569
5612
5654
5697
5739
5781
5824
43-48
2.36
0.8 5866
5909
5951
5993
6036
6078
6120
6162
6205
6247
42
2.37
0.8 6289
6331
6373
6415
6458
6500
6542
6584
6626
6668
42
2.38
0.8 6710
6752
6794
6836
6878
6920
6962
7004
7046
7087
42
2.39
0.8 7129
7171
7213
7255
7297
7338
7380
7422
7464
7505
42
2.40
0.8 7547
7589
7630
7672
7713
7755
7797
7838
7880
7921
42
2.41
0.8 7963
8004
8046
8087
8129
8170
8211
8253
8294
8335
41
2.42
0.8 8377
8418
8459
8501
8542
8583
8624
8666
8707
8748
41
2.43
0.8 8789
8830
8871
8913
8954
8995
9036
9077
9118
9159
41
2.44
0.8 9200
9241
9282
9323
9364
9405
9445
9486
9527
9568
41
2.45
0.8 9609
9650
9690
9731
9772
9813
9853
9894
9935
9975
41
2.46
0.9 0016
0057
0097
0138
0179
0219
0260
0300
0341
0381
41-40
2.47
0.9 0422
0462
0503
0543
0584
0624
0664
0705
0745
0786
40
2.48
0.9 0826
0866
0906
0947
0987
1027
1067
1108
1148
1188
40
2.49
0.9 1228
1268
1309
1349
1389
1429
1469
1509
1549
1589
40
2.60
0.91629
1669
1709
1749
1789
1829
1869
1909
1949
1988
40
J
1
2
3
4
5
6
7
8
9
I
TABLES.
133
Five-Place Natural Logarithms.
No.
1
2
3
4
5
6
7
8
9
D.
2.50
0.91629
1669
1709
1749
1789
1829
1869
1909
1949
1988
40
2.51
0.9 2028
2068
2108
2148
2188
2227
2267
2307
2346
2386
40
2.52
0.9 2426
2466
2505
2545
2584
2624
2664
2703
2743
2782
40
2.53
0.9 2S22
2S61
2901
2940
2980
3019
3059
3098
3138
3177
40-39
2.54
0.9 3216
3256
3295
3334
3374
3413
3452
3492
3531
3570
39
2.55
0.9 3609
3649
3688
3727
3766
3805
3844
3883
3923
3962
39
2.56
0.9 4001
4040
4079
4118
4157
4196
4235
4274
4313
4352
39
2.57
0.9 4391
4429
4468
4507
4546
4585
4624
4663
4701
4740
39
2.58
0.9 4779
4818
4856
4895
4934
4973
5011
5050
5089
5127
39
2.59
0.9 5166
5204
5243
5282
5320
5359
5397
5436
5474
5513
39-38
2.60
0.9 5551
5590
5628
5666
5705
5743
5782
5820
5858
5897
38
2.61
0.9 5935
5973
6012
6050
6088
6126
6165
6203
6241
6279
38
2.62
0.9 6317
6356
6394
6432
6470
6508
6546
6584
6622
6660
38
2.63
0.9 6698
6736
6774
6812
6S50
6S88
6926
6964
7002
7040
38
2.64
0.9 7078
7116
7154
7191
7229
7267
7305
7343
7380
7418
38
2.65
0.9 7456
7494
7531
7569
7607
7644
7682
7720
7757
7795
38
2.66
0.9 7833
7S70
7908
7945
7983
8020
8058
8095
8133
8170
38-37
2.67
0.9 8208
8245
8283
8320
8358
8395
8432
8470
8507
8544
37
2.68
0.9 8582
8619
8656
8694
8731
8768
8805
8843
8880
8917
37
2.69
0.9 8954
8991
9028
9066
9103
9140
9177
9214
9251
9288
37
2.70
0.9 9325
9362
9399
9436
9473
9510
9547
9584
9621
9658
37
2.71
0.9 9695
9732
9769
9806
9842
9879
9916
9953
9990
*0026
37
2.72
1.0 0063
0100
0137
0173
0210
0247
0284
0320
0357
0394
37
2.73
1.0 0430
0467
0503
0540
0577
0613
0650
0686
0723
0759
37
2.74
1.0 0796
0832
0869
0905
0942
0978
1015
1051
1087
1124
36
2.75
1.0 1160
1196
1233
1269
1305
1342
1378
1414
1451
1487
36
2.76
1.0 1523
1559
1596
1632
1668
1704
1740
1776
1813
1849
36
2.77
1.0 1S85
1921
1957
1993
2029
2065
2101
2137
2173
2209
36
2.78
1.0 2245
2281
2317
2353
2389
2425
2461
2497
2532
2588
36
2.79
1.0 2604
2640
2676
2712
2747
2783
2819
2855
2890
2926
36
2.80
1.0 2962
2998
3033
3069
3105
3140
3176
3212
3247
3283
36
2.81
1.0 3318
3354
3390
3425
3461
3496
3532
3567
3603
3638
36-35
2.82
1.0 3674
3709
3745
3780
3815
3851
3886
3922
3957
3992
35
2.S3
1.0 4028
4063
4098
4134
4169
4204
4239
4275
4310
4345
35
2.84
1.0 43 SO
4416
4451
4486
4521
4556
4591
4627
4662
4697
35
2.85
1.0 4732
4767
4802
4837
4872
4907
4942
4977
5012
5047
35
2.86
1.0 5082
5117
5152
5187
5222
5257
5292
5327
5361
5396
35
2.87
1.0 5431
5466
5501
5536
5570
5605
5640
5675
5710
5744
35
2.88
1.0 5779
5814
5848
5883
5918
5952
5987
6022
6056
6091
35
2.89
1.0 6126
6160
6195
6229
6264
6299
6333
6368
6402
6437
35-34
2.90
1.0 6471
6506
6540
6574
6609
6643
6678
6712
6747
6781
34
2.91
1.0 6815
6850
6884
6918
6953
6987
7021
7056
7090
7124
34
2.92
1.0 7158
7193
7227
7261
7295
7329
7364
7398
7432
7466
34
2.93
1.0 7500
7534
7568
7603
7637
7671
7705
7739
7773
7807
34
2.94
1.0 7841
7875
7909
7943
7977
8011
8045
8079
8113
8147
34
2.95
1.0 8181
8214
8248
8282
8316
8350
8384
8418
8451
8485
34
2.96
1.0 8519
8553
8586
8620
8654
8688
8721
8755
8789
8823
34
2.97
1.0 8856
8890
8924
8957
8991
9024
9058
9092
9125
9159
34
2.98
1.0 9192
9226
9259
9293
9326
9360
9393
9427
9460
9494
34-33
2.99
1.0 9527
9561
9594
9628
9661
9694
9728
9761
9795
9828
33
3.00
1.0 9861
9895
9928
9961
9994
*0028
0061
0094
0128
0161
33
1
2
3
4
5
6
7
8
9
134
TABLES.
Five-Place Natural Logarithms.
No,
1
2
3
4
5
6
7
8
9
D.
3.00
1.0 9861
9895
9928
9961
9994
*0028
0061
0094
0128
0161
33
3.01
1 10194
0227
0260
0294
0327
0360
0393
0426
0459
0493
33
3.02
i.l 0526
0559
0592
0625
0658
0691
0724
0757
0790
0823
33
3.03
1.1 0856
0889
0922
0955
0988
1021
1054
1087
1120
1153
33
3.04
1.1 1186
1219
1252
1284
1317
1350
1383
1416
1449
1481
33
3.05
1.11514
1547
1580
1612
1645
1678
1711
1743
1776
1809
33
3.06
1.1 1841
1874
1907
1939
1972
2005
2037
2070
2103
2135
33
3.07
1.1 2168
2200
2233
2265
2298
2330
2363
2396
2428
2460
33-33
3.08
1.1 2493
2525
2558
2590
2623
2655
2688
2720
2752
2785
32
3.09
1.12817
2849
2882
2914
2946
2979
3011
3043
3076
3108
32
8.10
1.13140
3172
3205
3237
3269
3301
3334
3366
3398
3430
32
3.11
1.1 3462
3494
3527
3559
3591
3623
3655
3687
3719
3751
32
3.12
1.1 3783
3815
3847
3879
3911
3943
3955
4007
4039
4071
32
3.13
1.1 4103
4135
4167
4199
4231
4263
4295
4327
4359
4390
32
3.14
1.14422
4454
4486
4518
4550
4581
4613
4645
4677
4708
32
3.15
1.1 4740
4772
4804
4835
4867
4899
4931
4962
4994
5026
32
3.16
1.1 5057
5089
5120
5152
5184
5215
5247
5278
5310
5342
32
3.17
1.15373
5405
5436
5468
5499
5531
5562
5594
5625
5657
32-31
3.18
1.1 5688
5720
5751
5782
5814
5845
5877
5908
5939
5971
31
3.19
1.1 6002
6033
6065
6096
6127
6159
6190
6221
6253
6284
31
3.20
1.1 6315
6346
6378
6409
6440
6471
6502
6534
6565
6596
31
3.21
1.1 6627
6658
6689
6721
6752
6783
6814
6845
6876
6907
31
3.22
1.1 6938
6969
7000
7031
7062
7093
7124
7155
7186
7217
31
3.23
1.1 7248
7279
7310
7341
7372
7403
7434
7465
7496
7526
31
3.24
1.1 7557
7588
7619
7650
7681
7712
7742
7773
7804
7835
31
3.25
1.1 7865
7896
7927
7958
7989
8019
8050
8081
8111
8142
31
3.26
1.18173
8203
8234
8265
8295
8326
8357
8387
8418
8448
31
3.27
1.1 8479
8510
8540
8571
8601
8632
8662
8693
8723
8754
31-30
3.28
1.1 8784
8815
8845
8876
8906
8937
8967
8998
9028
9058
30
3.29
1.1 9089
9119
9150
9180
9210
9241
9271
9301
9332
9362
30
3.30
1.1 9392
9423
9453
9483
9513
9544
9574
9604
9634
9665
30
3.31
1.1 9695
9725
9755
9785
9816
9846
9876
9906
9936
9966
30
3.32
1.1 9996
*0027
0057
0087
0117
0147
0177
0207
0237
0267
30
3.33
1.2 0297
0327
0357
0387
0417
0447
0477
0507
0537
0567
30
3.34
1.2 0597
0627
0657
06S7
0717
0747
0777
0806
0836
0866
30
3.35
1.2 0896
0926
0956
0986
1015
1045
1075
1105
1135
1164
30
3.36
1.21194
1224
1254
1283
1313
1343
1373
1402
1432
1462
30
3.37
1.2 1491
1521
1551
1580
1610
1640
1669
1699
1728
1758
30
3.38
1.2 1788
1817
1847
1876
1906
1935
1965
1994
2024
2053
30
3.39
1.2 2083
2112
2142
2171
2201
2230
2260
2289
2319
2348
29
3.40
1.2 2378
2407
2436
2466
2495
2524
2554
2583
2613
2642
29
3.41
1.2 2671
2701
2730
2759
2788
2818
2847
2876
2906
2935
29
3.42
1.2 2964
2993
3023
3052
3081
3110
3139
3169
3198
3227
29
3.43
1.2 3256
3285
3314
3343
3373
3402
3431
3460
3489
3518
29
3.44
1.2 3547
3576
3605
3634
3663
3692
3721
3750
3779
3808
29
3.45
1.2 3837
3866
3895
3924
3953
3982
4011
4040
4069
4098
29
3.46
1.2 4127
4156
4185
4214
4242
4271
4300
4329
4358
4387
29
3.47
1.2 4415
4444
4473
4502
4531
4559
4588
4617
4646
4674
29
3.48
1.2 4703
4732
4761
4789
4818
4847
4875
4904
4933
4962
29
3.49
1.2 4990
5019
5047
5076
5105
5133
5162
5191
5219
5248
29
8.50
1.2 5276
5305
5333
5362
5391
5419
5448
5476
5505
5533
29-28
1
2
3
4
5
6
7
8
9
TABLES.
Five-Place Natural Logarithms.
135
No.
1
2
3
4
5
6
7
8
9
D.
3.50
1.2 5276
5305
5333
5362
5391
5419
5448
5476
5505
5533
29-28
3.51
1.2 5562
5590
5619
5647
5675
5704
5732
5761
5789
5818
28
3.52
1.2 5846
5875
5903
5931
5960
5988
6016
6045
6073
6101
28
3.53
1.2 6130
6158
6186
6215
6243
6271
6300
6328
6356
6384
28
3.54
1.2 6413
6441
6469
6497
6526
6554
6582
6610
6638
6667
28
3.55
1.2 6695
6723
6751
6779
6807
6836
6864
6892
6920
6948
28
3.56
1.2 6976
7004
7032
7060
7088
7116
7144
7172
7201
7229
28
3.57
1.2 7257
7285
7313
7341
7369
7397
7424
7452
7480
7508
28
3.5S
1.2 7536
7564
7592
7620
7648
7676
7704
7732
7759
7787
28
3.59
1.2 7815
7843
7871
7899
7927
7954
7982
8010
8038
8066
28
3.60
1.2 8093
8121
8149
8177
8204
8232
8260
8288
8315
8343
28
3.61
1.2 8371
8398
8426
8454
8482
8509
8537
8564
8592
8620
28
3.62
1.2 8647
8675
8703
8730
8758
8785
8813
8841
8868
8896
28
3.63
1.2 8923
8951
8978
9006
9033
9061
9088
9116
9143
9171
28-27
3.64
1.2 9198
9226
9253
9281
9308
9336
9363
9390
9418
9445
27
3.65
1.2 9473
9500
9527
9555
9582
9610
9637
9664
9692
9719
27
3.66
1.2 9746
9774
9801
9828
9856
9883
9910
9937
9965
9992
27
3.67
1.3 0019
0046
0074
0101
0128
0155
0183
0210
0237
0264
27
3.68
1.3 0291
0318
0346
0373
0400
0427
0454
0481
0508
0536
27
3.69
1.3 0563
0590
0617
0644
0671
0698
0725
0752
0779
0806
27
3.70
1.3 0833
0860
0887
0914
0941
0968
0995
1022
1049
1076
27
3.71
1.3 1103
1130
1157
1184
1211
1238
1265
1292
1319
1345
27
3.72
1.3 1372
1399
1426
1453
1480
1507
1534
1560
1587
1614
27
3.73
1.3 1641
1668
1694
1721
1748
1775
1802
1828
1855
1882
27
3.74
1.3 1909
1935
1962
1989
2015
2042
2069
2096
2122
2149
27
3.75
1.3 2176
2202
2229
2256
2282
2309
2335
2362
2389
2415
27
3.76
1.3 2442
2468
2495
2522
2548
2575
2601
2628
2654
2681
27
3.77
1.3 2708
2734
2761
2787
2814
2840
2867
2893
2919
2946
27-26
3.78
1.3 2972
2999
3025
3052
3078
3105
3131
3157
3184
3210
26
3.79
1.3 3237
3263
3289
3316
3342
3368
3395
3421
3447
3474
26
3.80
1.3 3500
3526
3553
3579
3605
3632
3658
3684
3710
3737
26
3.81
1.3 3763
3789
3815
3842
3868
3894
3920
3946
3973
3999
26
3.82
1.3 4025
4051
4077
4104
4130
4156
4182
4208
4234
4260
26
3.83
1.3 4286
4313
4339
4365
4391
4417
4443
4469
4495
4521
26
3.84
1.3 4547
4573
4599
4625
4651
4677
4703
4729
4755
4781
26
3.85
1.3 4807
4833
4859
4885
4911
4937
4963
4989
5015
5041
26
3.86
1.3 5067
5093
5119
5144
5170
5196
5222
5248
5274
5300
26
3.87
1.3 5325
5351
5377
5403
5429
5455
5480
5506
5532
5558
26
3.88
1.3 5584
5609
5635
5661
5687
5712
5738
5764
5789
5815
26
3.89
1.3 5841
5867
5892
5918
5944
5969
5995
6021
6046
6072
26
3.90
1.3 6098
6123
6149
6175
6200
6226
6251
6277
6303
6328
26
3.91
1.3 6354
6379
6405
6430
6456
6481
6507
6533
6558
6584
26
3.92
1.3 6609
6635
6660
6686
6711
6737
6762
6788
6813
6838
26-25
3.93
1.3 6864
6889
6915
6940
6966
6991
7016
7042
7067
7093
25
3.94
1.3 7118
7143
7169
7194
7220
7245
7270
7296
7321
7346
25
3.95
1.3 7372
7397
7422
7447
7473
7498
7523
7549
7574
7599
25
3.96
1.3 7624
7650
7675
7700
7725
7751
7776
7801
7826
7851
25
3.97
1.3 7877
7902
7927
7952
7977
8002
8028
8053
8078
8103
25
3.98
1.3 8128
81-13
8178
8204
8229
8254
8279
8304
8329
8354
25
3.99
1.3 8379
8404
8429
8454
8479
8504
8529
8554
8579
8604
25
4.00
1.3 8629
8654
8679
8704
8729
8754
8779
8804
8829
8854
25
1
2
3
4
5
6
7
8
9
136
TABLES.
Five-Place Natural Logarithms.
No.
1
2
3
4
5
6
7
8
9
D.
4.00
1.3 8629
8654
8679
8704
8729
8754
8779
8804
8829
8854
25
4.01
1.3 8879
8904
8929
8954
8979
9004
9029
9054
9078
9103
25
4.02
1.3 9128
9153
9178
9203
9228
9252
9277
9302
9327
9352
25
4.03
1.3 9377
9401
9426
9451
9476
9501
9525
9550
9575
9600
25
4.04
1.3 9624
9649
9674
9699
9723
9748
9773
9798
9822
9847
25
4.0.S
1.3 9872
9896
9921
9946
9970
9995
*0020
0044
0069
0094
25
4.06
1.4 0118
0143
0168
0192
0217
0241
0266
0291
0315
0340
25
4.07
1.4 0364
0389
0413
0438
04b3
0487
0512
0536
0561
0585
25
4.08
1.4 0610
0634
0659
0683
0708
0732
0757
0781
0806
0830
25-24
4.09
1.4 0854
0879
0903
0928
0952
0977
1001
1025
1050
1074
24
4.10
1.4 1099
1123
1147
1172
1196
1221
1245
1269
1294
1318
24
4.11
1.4 1342
1367
1391
1415
1440
1464
1488
1512
1537
1561
24
4.12
1.4 1585
1610
1634
1658
1682
1707
1731
1755
1779
1804
24
4.13
1.4 1828
1852
1876
1900
1925
1949
1973
1997
2021
2045
24
4.14
1.4 2070
2094
2118
2142
2166
2190
2214
2239
2263
2287
24
4.15
1.4 2311
2335
2359
2383
2407
2431
2455
2479
2503
2527
24
4.16
1.4 2552
2576
2600
2624
2648
2672
2696
2720
2744
2768
24
4.17
1.4 2792
2816
2840
2864
2887
2911
2935
2959
2983
3007
24
4.18
1.4 3031
3055
3079
3103
3127
3151
3175
3198
3222
3246
24
4.19
1.4 3270
3294
3318
3342
3365
3389
3413
3437
3461
3485
24
4.20
1.4 3508
3532
3556
3580
3604
3627
3651
3675
3699
3723
24
4.21
1.4 3746
3770
3794
3817
3841
3865
3889
3912
3936
3960
24
4.22
1.4 3984
4007
4031
4055
4078
4102
4126
4149
4173
4197
24
4.23
1.4 4220
4244
4267
4291
4315
4338
4362
4386
4409
4433
24
4.24
1.4 4456
4480
4503
4527
4551
4574
4598
4621
4645
4668
24
4.25
1.4 4692
4715
4739
4762
4786
4809
4833
4856
4880
4903
24-23
4.26
1.4 4927
4950
4974
4997
5021
5044
5068
5091
5115
5138
23
4.27
1.4 5161
5185
5208
5232
5255
5278
5302
5325
5349
5372
23
4.28
1.4 5395
5419
5442
5465
5489
5512
5535
5559
5582
5605
23
4.29
1.4 5629
5652
5675
5699
5722
5745
5768
5792
5815
5838
23
4.30
1.4 5862
5885
5908
5931
5954
5978
6001
6024
6047
6071
23
4.31
1.4 6094
6117
6140
6163
6187
6210
6233
6256
6279
6302
23
4.32
1.4 6326
6349
6372
6395
6418
6441
6464
6487
6511
6534
23
4.33
1.4 6557
6580
6603
6626
6649
6672
6695
6718
6741
6764
23
4.34
1.4 6787
6810
6834
6857
6880
6903
6926
6949
6972
6995
23
4.35
1.4 7018
7041
7064
7087
7109
7132
7155
7178
7201
7224
23
4.36
1.4 7247
7270
7293
7316
7339
7362
7385
7408
7431
7453
23
4.37
1.4 7476
7499
7522
7545
7568
7591
7614
7636
7659
7682
23
4.38
1.4 7705
7728
7751
7773
7796
7819
7842
7865
7887
7910
23
4.39
1.4 7933
7956
7978
8001
8024
8047
8070
8092
8115
8138
23
4.40
1.4 8160
8183
8206
8229
8251
8274
8297
8319
8342
8365
23
4.41
1.4 8387
8410
8433
8455
8478
8501
8523
8546
8569
8591
23
4.42
1.4 8614
8637
8659
8682
8704
8727
8750
8772
8795
8817
23
4.43
1.4 8840
8863
8885
8908
8930
8953
8975
8998
9020
9043
23
4.44
1.4 9065
9088
9110
9133
9155
9178
9200
9223
9245
9268
23
4.45
1.4 9290
9313
9335
9358
9380
9403
9425
9448
9470
9492
23-22
4.46
1.4 9515
9537
9560
9582
9605
9627
9649
9672
9694
9716
22
4.47
1.4 9739
9761
9784
9806
9828
9851
9873
9895
9918
9940
22
4.48
1.4 9962
9985
*0007
0029
0052
0074
0096
0118
0141
0163
22
4.49
1.5 0185
0208
0230
0252
0274
0297
0319
0341
0363
0386
22
4.50
1.5 0408
0430
0452
0474
0497
0519
0541
0563
0585
0608
22
1
2
3
4
5
6
7
8
9
TABLES.
Five-Place Natural Logarithms.
137
No.
1
2
3
4
5
6
7
8
9
D.
4.50
1.5 0408
0430
0452
0474
0497
0519
0541
0563
0585
0608
22
4.51
1.5 0630
0652
0674
0696
0718
0741
0763
0785
0807
0829
22
4.52
1.5 0851
0873
0895
0918
0940
0962
0984
1006
1028
1050
22
4.53
1.5 1072
1094
1116
1138
1160
1183
1205
1227
1249
1271
22
4.54
1.5 1293
1315
1337
1359
1381
1403
1425
1447
1469
1491
22
4.55
1.5 1513
1535
1557
1579
1601
1623
1645
1666
1688
1710
22
4.56
1.5 1732
1754
1776
1798
1820
1842
1864
1886
1908
1929
22
4.57
1.5 1951
1973
1995
2017
2039
2061
2083
2104
2126
2148
22
4.58
1.5 2170
2192
2214
2235
2257
2279
2301
2323
2344
2366
22
4.59
1.5 2388
2410
2432
2453
2475
2497
2519
2540
2562
2584
22
4.60
1.5 2606
2627
2649
2671
2693
2714
2736
2758
2779
2801
22
4.61
1.5 2823
2844
2866
2888
2910
2931
2953
2975
2996
3018
22
4.62
1.5 3039
3061
3083
3104
3126
3148
3169
3191
3212
3234
22
4.63
1.5 3256
3277
3299
3320
3342
3364
3385
3407
3428
3450
22
4.64
1.5 3471
3493
3515
3536
3558
3579
3601
3622
3644
3665
22
4.65
1.5 3687
3708
3730
3751
3773
3794
3816
3837
3859
3880
22-21
4.66
1.5 3902
3923
3944
3966
3987
4009
4030
4052
4073
4094
21
4.67
1.5 4116
4137
4159
4180
4202
4223
4244
4266
4287
4308
21
4.68
1.5 4330
4351
4373
4394
4415
4437
4458
4479
4501
4522
21
4.69
1.5 4543
4565
4586
4607
4629
4650
4671
4692
4714
4735
21
4.70
1.5 4756
4778
4799
4820
4841
4863
4884
4905
4926
4948
21
4.71
1.5 4969
4990
5011
5032
5054
5075
5096
5117
5138
5160
21
4.72
1.5 5181
5202
5223
5244
5266
5287
5308
5329
5350
5371
21
4.73
1.5 5393
5414
5435
5456
5477
5498
5519
5540
5562
5583
21
4.74
1.5 5604
5625
5646
5667
5688
5709
5730
5751
5772
5793
21
4.75
1.5 5814
5836
5857
5878
5899
5920
5941
5962
5983
6004
21
4.76
1.5 6025
6046
6067
6088
6109
6130
6151
6172
6193
6214
21
4.77
1.5 6235
6256
6277
6298
6318
6339
6360
6381
6402
6423
21
4.78
1.5 6444
6465
6486
6507
6528
6549
6569
6590
6611
6632
21
4.79
1.5 6653
6674
6695
6716
6737
6757
6778
6799
6820
6841
21
4.80
1.5 6862
6882
6903
6924
6945
6966
6987
7007
7028
7049
21
4.81
1.5 7070
7090
7111
7132
7153
7174
7194
7215
7236
7257
21
4.82
1.5 7277
7298
7319
7340
7360
7381
7402
7423
7443
7464
21
4.83
1.5 7485
7505
7526
7547
7567
7588
7609
7629
7650
7671
21
4.84
1.5 7691
7712
7733
7753
7774
7795
7815
7836
7857
7877
21
4.85
1.5 7898
7918
7939
7960
7980
8001
8022
8042
8063
8083
21
4.86
1.5 8104
8124
8145
8166
8186
8207
8227
8248
8268
8289
21
4.87
1.5 8309
8330
8350
8371
8391
8412
8433
8453
8474
8494
21-20
4.88
1.5 8515
8535
8555
8576
8596
8617
8637
8658
8678
8699
20
4.89
1.5 8719
8740
8760
8781
8S01
8821
8842
8862
8883
8903
20
4.90
1.5 8924
8944
8964
8985
9005
9026
9046
9066
9087
9107
20
4.91
1.5 9127
9148
9168
9188
9209
9229
9250
9270
9290
9311
20
4.92
1.5 9331
9351
9371
9392
9412
9432
9453
9473
9493
9514
20
4.93
1.5 9534
955'
9574
9595
9615
9635
9656
9676
9696
9716
20
4.94
1.5 9737
9757
9777
9797
9817
9838
9858
9878
9898
9919
20
4.95
1.5 9939
9959
9979
9999 *0020
0040
0060
OOSO
0100
0120
20
4.96
1.6 0141
0161
0181
0201
0221
0241
0261
028?
0302
0322
20
4.97
1.6 0342
0362
0382
0402
0422
0443
0463
0483
0503
0523
20
4.98
1.6 0543
0563
0583
0603
0623
0643
0663
0683
0704
0724
20
4.99
1.6 0744
0764
0784
0804
0824
0844
0864
0884
0904
0924
20
5>00
1.6 0944
0964
0984
1004
1024
1044
1064
1084
1104
1124
20
1
2
3
4
5
6
7
8
9
138
TABLES.
Five-Place Natural Logarithms.
No.
1
2
3
4
5
6
7
8
9
D.
5.0
1.6 0944
1144
1343
1542
1741
1939
2137
2334
2531
2728
200-196
5.1
1.6 2924
3120
3315
3511
3705
3900
4094
4287
44S1
4673
196-192
5.2
1.6 4866
5058
5250
5441
5632
5823
6013
6203
6393
6582
192-189
5.3
1.6 6771
6959
7147
7335
7523
7710
7896
8083
8269
8455
189-185
5.4
1.6 8640
8825
9010
9194
9378
9562
9745
9928 *0111
0293
185-182
5.5
1.7 0475
0656
0838
1019
1199
1380
1560
1740
1919
2098
182-179
5.6
1.7 2277
2455
2633
2811
2988
3166
3342
3519
3695
3871
178-176
5.7
1.7 4047
4222
4397
4572
4746
4920
5094
5267
5440
5613
175-173
5.8
1.7 5786
5958
6130
6302
6473
6644
6815
6985
7156
7326
172-170
5.9
1.7 7495
7665
7834
8002
8171
8339
8507
8675
8842
9009
169-167
6.0
1.7 9176
9342
9509
9675
9840
*0006
0171
0336
0500
0665
167-164
6.1
1.8 0829
0993
1156
1319
1482
1645
1808
1970
2132
2294
164-161
6.2
1.8 2455
2616
2777
2938
3098
3258
3418
3578
3737
3896
161-159
6.3
1.8 4055
4214
4372
4530
4688
4845
5003
5160
5317
5473
159-156
6.4
1.8 5630
5786
5942
6097
6253
6408
6563
6718
6872
7026
156-154
6.5
1.8 7180
7334
7487
7641
7794
7947
8099
8251
8403
8555
154-152
6.6
1.8 8707
8858
9010
9160
931]
9462
9612
9762
9912 *0061
151-149
6.7
1.9 0211
0360
0509
0658
0806
0954
1102
1250
1398
1545
149-147
6.8
1.9 1692
1839
1986
2132
2279
2425
2571
2716
2862
3007
147-145
6.9
1.9 3152
3297
3442
3586
3730
3874
4018
4162
4305
4448
145-143
7.0
1.9 4591
4734
4876
5019
5161
5303
5445
5586
5727
5869
143-141
7.1
1.9 6009
6150
6291
6431
6571
6711
6851
6991
7130
7269
141-139
7.2
1.9 7408
7547
7685
7824
7962
8100
8238
8376
8513
8650
139-137
7.3
1.9 8787
8924
9061
9198
9334
9470
9606
9742
9877 *0013
137-135
7.4
2.0 0148
0283
0418
0553
0687
0821
0956
1089
1223
1357
135-133
7.5
2.0 1490
1624
1757
1890
2022
2155
2287
2419
2551
2683
133-132
7.6
2.0 2815
2946
3078
3209
3340
3471
3601
3732
3862
3992
131-130
7.7
2.0 4122
4252
4381
4511
4640
4769
4898
5027
5156
5284
130-128
7.8
2.0 5412
5540
5668
5796
5924
6051
6179
6306
6433
6560
128-127
7.9
2.0 6686
6813
6939
7065
7191
7317
7443
7568
7694
7819
127-125
8.0
2.0 7944
8069
8194
8318
8443
8567
8691
8815
8939
9063
125-124
8.1
2.0 9186
9310
9433
9556
9679
9802
9924 *0047
0169
0291
123-122
8.2
2.1 0413
0535
0657
0779
0900
1021
1142
1263
1384
1505
122-121
8.3
2.1 1626
1746
1866
1986
2106
2226
2346
2465
2585
2704
120-119
8.4
2.1 2823
2942
3061
3180
3298
3417
3535
3653
3771
3889
119-118
8.5
2.1 4007
4124
4242
4359
4476
4593
4710
4827
4943
5060
118-116
8.6
2.1 5176
5292
5409
5524
5640
5756
5871
5987
6102
6217
116-115
8.7
2.1 6332
6447
6562
6677
6791
6905
7020
7134
7248
7361
115-114
8.8
2.1 7475
7589
7702
7816
7929
8042
8155
8267
8380
8493
114-112
8.9
2.1 8605
2.1 9722
8717
9834
8830 8942
9944 *0055
9054
0166
9165
9277
9389
9500
9611
112-111
9.0
0276
0387
0497
0607
0717
111-110
9.1
2.2 0827
0937
1047
1157
1266
1375
1485
1594
1703
1812
110-109
9.2
2.2 1920
2029
2138
2246
2354
2462
2570
2678
2786
2894
109-108
9.3
2.2 3001
3109
3216
3324
3431
3538
3645
3751
3858
3965
107-106
9.4
2.2 4071
4177
4284
4390
4496
4601
4707
4813
4918
5024
106-105
9.5
2.2 5129
5234
5339
5444
5549
5654
5759
5863
5968
6072
105-104
9.6
2.2 6176
6280
6384
6488
6592
6696
6799
6903
7006
7109
104-103
9.7
2.2 7213
7316
7419
7521
7624
7727
7829
7932
8034
8136
103-102
9.8
2.2 8238
8340
8442
8544
8646
8747
8849
8950
9051
9152
102-101
9.9
2.2 9253
9354
9455
9556
9657
9757
9858
9958 *0058
0158
101-100
10.0
2.3 0259
0358
0458
0558
0658
0757
0857
0956
1055
1154
100-99
1
2
3
4
5
6
7
8
9
TABLES. 139
The Natural Logarithms (each increased by 10.) of Numbers between 0.00 and 0.99.
No.
1
2
3
4
5
6
7
8
9
0.0
5.395
6.088
6.493
6.781
7.004
7.187
7.341
7.474
7.592
0.1
7.697
7.793
1880
7.960
8.034
8.103
8.167
8.228
8.285
8.339
0.2
8.391
8.439
8.486
8.530
8.573
8.614
8.653
8.691
8.727
8.762
0.3
8.796
8.829
8.861
8.891
8.921
8.950
8.978
9.006
9.032
9.058
0.4
9.084
9.10S
9.132
9.156
9.179
9.201
9.223
9.245
9.266
9.287
0.5
9.307
9.327
9.346
9.365
9.384
9.402
9.420
9.438
9.455
9.472
0.6
9.489
9.506
9.522
9.538
9.554
9.569
9.584
9.600
9.614
9.629
0.7
9.643
9.658
9.671
9.685
9.699
9.712
9.726
9.739
9.752
9.764
0.8
9.777
9.789
9.802
9.814
9.826
9.837
9.849
9.861
9.872
9.883
0.9
9.895
9.906
9.917
9.927
9.938
9.949
9.959
9.970
9.980
9.990
Note : loggX = logioX • lege 10 =: (2.30259) logio x.
The Natural Logarithms of Whole Numbers from 10 to 209.
No.
1
2
3
4
5
6
7
8
9
1
2.3026
3979
4849
5649
6391
7080
7726
8332
8904
9444
2
2.9957
*0445
0910
1355
1781
2189
2581
2958
3322
3673
3
3.4012
4340
4657
4965
5264
5553
5835
6109
6376
6636
4
3.6889
7136
7377
7612
7842
8067
8286
8501
8712
8918
5
3.9120
9318
9512
9703
9890
*0073
0254
0431
0604
0775
6
4.0943
1109
1271
1431
1589
1744
1897
2047
2195
2341
7
4.2485
2627
2767
2905
3041
3175
3307
3438
3567
3694
8
4.3820
3944
4067
4188
4308
4427
4543
4659
4773
4886
9
4.4998
5109
5218
5326
5433
5539
5643
5747
5850
5951
10
4.6052
6151
6250
6347
6444
6540
6634
6728
6821
6913
11
4.7005
7095
7185
7274
7362
7449
7536
7622
7707
7791
12
4.7875
7958
8040
8122
8203
8283
8363
8442
8520
8598
13
4.8675
8752
8828
8903
8978
9053
9127
9200
9273
9345
14
4.9416
94SS
9558
9628
9698
9767
9836
9904
9972
*0039
15
5.0106
0173
0239
0304
0370
0434
0499
0562
0626
0689
16
5.0752
0814
0876
0938
0999
1059
1120
1180
1240
1299
17
5.1358
P17
1475
1533
1591
1648
1705
1762
1818
1874
18
5.1930
1985
2040
2095
2149
2204
2257
2311
2364
2417
19
5.2470
2523
2575
2627
2679
2730
2781
2832
2883
2933
20
5.2983
3033
3083
3132
3181
3230
3279
3327
3375
3423
Note : loge 10 = 2,30258509.
lege 100 = 4.60517019.
140
TABLES.
The Common Logarithms of r (n) for Values of n between 1 and 2.
r(n)= j x"-i-e-^dx= j log- dx.
n
9^
o
n
o
ho
O
r— »
n
3^
o
n
3;
2
0^
n
3;
bO
1.01
1.9975
1.21
T.9617
1.41
1.9478
1.61
1.9517
1.81
1.9704
1.02
1.9951
1.22
1.9605
1.42
1.9476
1.62
1.9523
1.82
1.9717
1.03
1.9928
1.23
1.9594
1.43
1.9475
1.63
1.9529
1.83
1.9730
1.04
1.9905
1.24
1.9583
1.44
1.9473
1.64
1.9536
1.84
1.9743
1.05
1.9883
1.25
1.9573
1.45
1.9473
1.65
1.9543
1.85
1.9757
1.06
T.9862
1.26
1.9564
1.46
1.9472
1.66
1.9550
1.86
1.9771
1.07
T.9841
1.27
T.9554
1.47
1.9473
1.67
1.9558
1.87
1.9786
l.OS
1.9821
1.28
1.9546
1.48
1.9473
1.68
T.9566
1.88
1.9800
1.09
1.9802
1.29
1.9538
1.49
1.9474
1.69
1.9575
1.89
1.9815
1.10
1.9783
1.30
1.9530
1.50
1.9475
1.70
T.9584
1.90
1.9831
1.11
1.9765
1.31
T.9523
1.51
1.9477
1.71
1.9593
1.91
T.9846
1.12
1.9748
1.32
1.9516
1.52
1.9479
1.72
1.9603
1.92
1.9862
1.13
1.9731
1.33
1.9510
1.53
1.9482
1.73
1.9613
1.93
1.9878
1.14
1.9715
1.34
T.9505
1.54
1.9485
1.74
T.9623
1.94
1.9895
1.15
1.9699
1.35
1.9500
1.55
1.9488
1.75
1.9633
1.95
1.9912
1.16
1.9684
1.36
1.9495
1.56
1.9492
1.76
1.9644
1.96
T.9929
1.17
1.9669
1.37
1.9491
1.57
1.9496
1.77
T.9656
1.97,
1.9946
1.18
1.9655
1.38
1.9487
1.5S
1.9501
1.78
T.9667
1.98
1.9964
1.19
1.9642
1.39
1.9483
1.59
1.9506
1.79
1.9679
1.99
1.9982
1.20
1.9629
1.40
1.9481
1.60
1.9511
1.80
1.9691
200
0.0000
r(2 + i) = z-r(2), z>i.
TABLES.
141
NATURAL TRIGONOMETRIC FUNCTIONS.
Angle.
Sin.
Csc.
Tan.
Ctn,
Sec.
Cos.
0°
0.000
00
0.000
00
1.000
1.000
90°
1
0.017
57.30
0.017
57.29
1.000
1.000
89
2
0.035
28.65
0.035
28.64
1.001
0.999
88
3
0.052
19.11
0.052
19.08
1.001
0.999
87
4
0.070
14.34
0.070
14.30
1.002
0.998
86
5°
0.0S7
11.47
0.0S7
11.43
1.004
0.996
85°
6
0.105
9.567
0.105
9.514
1.006
0.995
84
7
0.122
8.206
0.123
8.144
1.008
0.993
83
8
0.139
7.185
0.141
7.115
1.010
0.990
82
9
0.156
6.392
0.158
6.314
1.012
0.988
81
10°
0.174
5.759
0.176
5.671
1.015
0.985
80°
11
0.191
5.241
0.194
5.145
1.019
0.982
79
12
0.208
4.810
0.213
4.705
1.022
0.978
78
13
0.225
4.445
0.231
4.331
1.026
0.974
77
14
0.242
4.134
0.249
4.011
1.031
0.970
76
15°
0.259
3.864
0.268
3.732
1.035
0.966
75°
16
0.276
3.628
0.287
3.487
1.040
0.961
74
17
0.292
3.420
0.306
3.271
1.046
0.956
73
18
0.309
3.236
0.325
3.078
1.051
0.951
72
19
0.326
3.072
0.344
2.904
1.058
0.946
71
20°
0.342
2.924
0.364
2.747
1.064
0.940
70°
21
0.358
2.790
0.384
2.605
1.071
0.934
69
22
0.375
2.669
0.404
2.475
1.079
0.927
68
23
0.391
2.559
0.424
2.356
1.086
0.921
67
24
0.407
2.459
0.445
2.246
1.095
0.914
66
25^
0.423
2.366
0.466
2.145
1.103
0.906
65°
26
0.438
2.281
0.488
2.050
1.113
0.899
64
27
0.454
2.203
0.5 JO
1.963
1.122
0.891
63
28
0.469
2.130
0.532
1.881
1.133
0.883
62
29
0.485
2.063
0.554
1.804
1.143
0.875
61
30"
0.500
2.000
0.577
1.732
1.155
0.866
60°
31
0.515
1.942
0.601
1.664
1.167
0.857
59
32
0.530
1.8S7
0.625
1.600
1.179
0.848
58
33
0.545
1.836
0.649
1.540
1.192
0.839
57
34
0.559
1.788
0.675
1.483
1.206
0.829
56
35°
0.574
1.743
0.700
1.428
1.221
0.819
55°
36..
'37 ;^
S8
0.588
1.701
1 £.ilC%- ■■■
0.727
1.376
-1.327
1.280
1.236
1.252
1.269
0.809
0.799
0.788
54
53
52
0.602
0.616
1.662
1.624
0.754-
0.781
39
0.629
1.589
0.810
1.235
1.287
0.777
51
40°
0.643
1.556
0.839
1.192
1.305
0.766
50° '
41
0.656
1.524
0.869
1.150
1.325
0.755
49
42
0.669
1.494
0.900
1.111
1.346
0.743
48
43
0.682
1.466
0.933
1.072
1.367
0.731
47
44
0.695
1.440
0.966
1.036
1.390
0.719
46
45°
0.707
1.414
1.000
1.000
1.414
0.707
45°
Cos.
Sec.
Ctn.
Tan.
Csc.
Sin.
Angle.
142
TABLES.
Logarithms.
N
1
2
3
4 5
6
7
8
9
P.P.
1.2- 3. 4- 5
lO
0000
0043
0086
0128
0170
0212
0253
0294
0334
0374
4- 8-12.17.21
11
0414
0453
0492
0531
0569
0607
0645
0682
0719
0755
4 8.11.10-19
12
0792
0828
0864
0899
0934
0969
1004
1038
1072
1106
3- 7-10. 14-17
13
1139
1173
1206
1239
1271
1303
1335
1367
1399
1430
3- 610.13.16
14
15
1461
1492
1523
1553
1584
1614
1044
1673
1703
1732
3. 6. 9.12.15
1761
1790
1818
1847
1875
1903
1931
1959
1987
2014
3. 6. 8-11.14
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
3. 5. 81113
17
2304
2330
2355
2380
2405
2430
2455
2480
2504
2529
2. 5- 7-10-12
18
2553
2577
2601
2625
2648
2672
2695
2718
2742
2765
2- 5- 7. 9-12
19
20
2788
2810
2833
2856
2878
2900
2923
2945
2967
2989
2. 4. 7. 911
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
2- 4- 6- 8 11
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
2. 4. 6. 810
22
3424
3444
3464
3483
3502
3522
3541
3560
3579
3598
2- 4. 6- 8.10
23
3617
3636
3655
3674
3692
3711
3729
3747
3766
3784
2. 4. 5- 7- 9
24
25
3802
3820
3838
3856
3874
3892
3909
3927
3945
3962
2- 4. 5- 7. 9
3979
3997
4014
4031
4048
4065
4082
4099
4110
4133
2 3. 5. 7. 9
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
2. 3. 5. 7. 8
27
4314
4330
4346
4362
4378
4393
4409
4425
4440
4456
2. 3. 5. 6- 8
28
4472
4487
4502
4518
4533
4548
4564
4579
4594
4609
2- 3. 5. 6. 8
29
30
4624
4639
4654
4669
4683
4698
4713
4728
4742
4757
1. 3- 4. 6. 7
4771
4786
4800
4814
4829
4843
4857
4871
4886
4900
1- 3. 4. 6. 7
31
4914
4928
4942
4955
4969
4983
4997
5011
5024
5038
1- 3- 4. 6- 7
32
5051
5065
5079
5092
5105
5119
5132
5145
5159
5172
1. 3. 4- 5- 7
33
5185
5198
5211
5224
5237
5250
5263
5276
5289
5302
1- 3. 4- 5. 6
34
35
5315
5328
5340
5353
5366
5378
5391
5403
5416
5428
1. 3- 4- 5- 6
5441
5453
5465
5478
5490
5502
5514
5527
5539
5551
1- 2- 4- 5- 6
36
5563
5575
5587
5599
5611
5623
5635
5647
5658
5670
1- 2- 4. 5. 6
37
5682
5694
5705
5717
5729
5740
5752
5763
5775
5786
1-2. 3- 5. 6
38
5798
5809
5821
5832
58i3
5855
5866
5877
5888
5899
1- 2- 3. 5. 6
39
40
5911
5922
5933
5944
5955
5966
5977
5988
5999
6010
1- 2- 3. 4- 6
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
1- 2- 3. 4. 5
41
6128
6138
6149
6160
6170
6180
6191
6201
6212
6222
1- 2. 3. 4. 5
42
6232
6243
6253
6263
6274
6284
6294
6304
6314
6325
1- 2- 3- 4. 5
43
6335
6345
6355
6365
6375
6385
6395
6405
6415
6425
1- 2. 3. 4- 5
44
45
6435
6444
6454
6464
6474
6484
6493
6503
6513
6522
1. 2. 3- 4- 5
6532
6542
6551
65G1
6571
6580
6590
6599
6809
6618
1. 2- 3. 4. 5
46
6628
6637
6646
6656
6665
6675
6684
6693
6702
6712
1. 2- 3- 4- 5
47
6721
6730
6739
6749
6758
6767
6776
6785
6794
6803
1. 2. 3. 4. 5
48
6812
6821
6830
6839
6848
6857
6366
6875
6884
6893
1. 2. 3 4. 4
49
50
6902
6911
6920
6928
6937
6946
6955
6964
6972
6981
1- 2- 3- 4- 4
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
1- 2. 3. 3. 4
51
7076
7084
7093
7101
7110
7118
7126
7135
7143
7152
1- 2- 3. 3. 4
52
7160
7168
7177
7185
7193
7202
7210
7218
7226
7235
1- 2- 2- 3. 4
53
7243
7251
7259
7267
7275
7284
7292
7300
7308
7316
1- 2. 2- 3. 4
54
7324
7332
7340
7348
7356
7364
7372
7380
7388
7396
1. 2- 2. 3. 4
NoTK. — This page and the three that follow it are taken from the Mathematical
Tables of Prof. J. M. Peirce, published by ^Messrs. Ginn & Co.
TABLES.
Logarithms.
143
[
N
12 3
4 5
6
7
8
9
P P.
1. ^
• 3- 4. 5
55
7404
7412 7419 7427
7435
7443
7451
7459
7466
7474
J. 2
. 2. 3- 4
56
7482
7490 7497 7505
7513
7520
7528
7536
7543
7551
1-2
.23-4
57
7559
7566 7574 7582
7589
7597
7604
7612
7619
7627
1. 2
• 2. 3. 4
58
7634
7642 7649 7657
7664
7672
7679
7686
7694
7701
2-3. 4
59
60
7709
7716 7723 7731
7738
7745
7752
7760
7767
7774
2. 3 4
7782
7789 7796 7803
7810
7818
7825
7832
7839
7846
2-3.4
61
7853
7860 7868 7875
7882
7889
7896
7903
7910
7917
2.3-4
62
7924
7931 7938 7945
7952
7959
7966
7973
7980
7987
2.3.3
63
7993
8000 8007 8014
8021
8028
8035
8041
8048
8055
2.3- 3
64
65
8062
8069 8075 8082
8089
8096
8102
8109
8116
8122
2.3.3
8129
8136 8142 8149
8156
8162
8169
8176
8182
8189
2-3.3
66
8195
8202 8200 8215
8222
8228
8235
8241
8248
8254
2.3.3
67
8261
8267 8274 8280
8287
8293
8299
8306
8312
8319
2.3.3
68
8325
8331 8338 8344
8351
8357
8363
8370
8376
8382
2.3. 3
69
70
8388
8395 8401 8407
8414
8420
8426
8432
8439
8445
2. 3.3
8451
8457 8463 8470
8476
8482
8488
8494
8500
8506
2.2.3
71
8513
8519 8525 8531
8537
8543
8549
8555
8561
8567
2-2.3
72
8573
8579 8585 8591
8597
8603
8609
8615
8621
8627
2.2.3
73
8633
8639 8645 8651
8657
8663
8669
8675
8681
8686
2. 2-3
74
75
8692
8698 8704 8710
8716
8722
8727
8733
8739
8745
2-2.3
8751
8756 8762 8768
8774
8779
8785
8791
8797
8802
2.2.3
76
8808
8814 8820 8825
8831
8837
8842
8848
8854
8859
2-2-3
77
8865
8871 8876 8882
8887
8893
8899
8904
8910
8915
2.2.3
78
8921
8927 8932 8938
8943
8949
8954
8960
8965
8971
2.2.3
79
80
8976
8982 8987 8993
8998
9004
9009
9015
9020
9025
2- 2. 3
9031
9036 9042 9047
9053
9058
9063
9069
9074
9079
2-2.3
81
9085
9090 9096 9101
9106
9112
9117
9122
9128
9133
2.2.3
82
9138
9143 9149 9154
9159
9165
9170
9175
9180
9186
2.2-3
83
9191
9196 9201 9206
9212
9217
9222
9227
9232
9238
2.2-3
84
85
9243
9248 9253 9258
9263
9269
9274
9279
9284
9289
2.2.3
9294
9299 9304 9309
9315
9320
9325
9330
9335
9340
2 2.3
86
9345
9350 9355 9360
9365
9370
9375
9380
9385
9390
2-2.3
87
9395
9400 9405 9410
9415
9420
9425
9430
9435
9440
0. 1
1.2.2
88
9445
9450 9455 9460
9465
9469
9474
9479
9484
9489
0.1
1- 2-2
89
90
9494
9499 9504 9509
9513
9518
9523
9528
9533
9538
0. 1
1.2.2
9542
9547 9552 9557
9562
9566
9571
9576
9581
9586
0- 1
1.2.2
91
9590
9595 9600 9605
9609
9614
9619
9624
9628
9633
0. 1
1.2.2
92
9638
9643 9647 9652
9657
9661
9666
9671
9675
9680
0- 1
1.2-2
93
9685
9689 9694 9699
9703
9708
9713
9717
9722
9727
0- 1
1.2.2
94
95
9731
9736 9741 9745
9750
9754
9759
9763
9768
9773
1
1. 2- 2
9777
9782 9786 9791
9795
9800
9805
9809
9814
9818
0. 1
1.2.2
96
9823
9827 9832 9836
9841
9845
9850
9854
9859
9863
0. 1
1.2.2
97
9868
9872 9877 9881
9886
9890
9894
9899
9903
9908
0- 1
1.2.2
98
9912
9917 9921 9926
9930
9934
9939
9943
9948
9952
0- 1
. 1. 2. 2
99
9956
9961 9965 9969
9974
9978
9983
9987
9991
9996
0- 1
1.2-2
log !r= 0.49715-
log e = 0.43429 -
144
TABLES.
Logarithms.
N
T
2
3 4 5
6
7
8
9 10
100
0000
0004
0009
0013 0017
0022
0026
0030
0035
0039
0043
101
0043
0043
0052
0056 0060
0065
0069
0073
0077
0082
0086
102
0086
0090
0095
0099 0103
0107
0111
0116
0120
0124
0128
103
0128
0133
0137
0141 0145
0149
0154
0158
0162
0166
0170
104
105
0170
0175
0179
0183 0187
0191
0195
0199
0204
0208
0212
0212
0216
0220
0224 0228
0233
0237
0241
0245
0249
0253
106
0253
0257
0261
0265 0269
0273
0278
0282
0286
0290
0294
107
0294
0298
0302
0306 0310
0314
0318
0322
0326
0330
0334
108
0334
0338
0342
0346 0350
0354
0358
0362
0366
0370
0374
109
110
0374
0378
0382
0386 0390
0394
0398
0402
0406
0410
0414
0414
0418
0422
0426 0430
0434
0438
0441
0445
0449
0453
111
0453
0457
0461
0465 0469
0473
0477
0481
0484
0488
0492
112
0492
0496
0500
0504 0508
0512
0515
0519
0523
0527
0531
113
0531
0535
0538
0542 0546
0550
0554
0558
0561
0565
0569
114
115
0569
0573
0577
0580 0584
0588
0592
0596
0599
0603
0607
0607
0611
0615
0618 0622
0626
0630
0633
0637
0641
0645
116
0645
0648
0652
0656 0660
0663
0667
0671
0674
0678
0682
117
0682
0686
0689
0693 0697
0700
0704
0708
0711
0715
0719
118
0719
0722
0726
0730 0734
0737
0741
0745
0748
0752 1 0755
119
120
0755
0759
0763
0766 0770
0774
0777
0781
0785
0788 \ 0792
0792
0795
0799
0803 0806
0810
0813
0817
0821
0824
0828
121
0828
0831
0835
0839 0842 ' 0846
0849
0853
0856
0860
0864
122
0864
0867
0871
0874 0878 / 0881
0885
0888
0892
0896
0899
123
0899
0903
0906
0910 0913
0917
0920
0924
0927
0931
0934
124
125
0934
0938
0941
0945 0948
0952
0955
0959
0962
0966
0969
0969
0973
0976
0980 0983
0986
0990
0993
0997
1000
1004
126
1004
1007
1011
1014 1017
1021
1024
1028
1031
1035
1038
127
1038
1041
1045
1048 1052
1055
1059
1062
1065
1069
1072
128
1072
1075
1079
1082 1086
1089
1092
1096
1099
1103
1106
129
130
1106
1109
1113
1116 1119
1123
1126
1129
1133
1136
1139
1139
1143
1146
1149 1153
1156
1159
1163
1166
1169
1173
131
1173
1176
1179
1183 1186
1189
1193
1196
1199
1202
1206
132
1208
1209
1212
1216 1219
1222
1225
1229
1232
1235
1239
133
1239
1242
1245
1248 1252
1255
1258
1261
1265
1268
1271
134
135
1271
1274
1278
1281 1284
1287
1290
1294
1297
1300
1303
1303
1307
1310
1313 1316
1319
1323
1326
1329
1332
1335
136
1335
1339
1342
1345 1348
1351
1355
1358
1361
1364
1367
137
1367
1370
1374
1377 1380
1383
1386
1389
1392
1396
1399
138
1399
1402
1405
1408 1411
1414
1418
1421
1424
1427
1430
139
140
1430
1433
1436
1440 1443
1446
1449
1452
1455
1458
1461
1461
1464
1467
1471 1474
1477
1480
1483
1486
1489
1492
141
1492
1495
1498
1501 1504
1508
1511
1514
1517
1520
1523
142
1523
1526
1529
1532 1535
1538
1541
1544
1547
1550
1553
143
1553
1556
1559
1562 1565
1569
1572
1575
1578
1581
1584
144
145
1584
1587
1590
1593 1596
1599
1602
1605
1608
1611
1614
1614
1617
1620
1623 1626
1629
1632
1635
1638
1641
1644
146
1644
1647
1649
1652 1655
1658
1661
1664
1667
1670
1673
147
1673
1676
1679
1682 1685
1688
1691
1694
1697
1700
1703
148
1703
1706
1708
1711 1714
1717
1720
1723
1726
1729
1732
149
1732
1735
1738
1741 1744
1746
1749
1752
1755
1758
1761
TABLES.
Logarithms.
145
N
1 2
3
4 5
6
7 8
9
10
150
1761
1764 1767
1770
1772
1775
1778
1781 1784
1787
1790
151
1790
1793 1796
1798
1801
1804
1807
1810 1813
1816
1818
152
1818
1821 1824
1827
1830
1833
1836
1838 1841
1844
1847
153
1847
1850 1853
1855
1858
1861
1864
1867 1870
1872
1875
154
155
1875
1878 1881
1884
1886
1889
1892
1895 1898
1901
1903
1903
1906 1909
1912
1915
1917
1920
1923 1926
1928
1931
156
1931
1934 1937
1940
1942
1945
1948
1951 1953
1956
1959
157
1959
1962 1965
1967
1970
1973
1976
1978 1981
1984
1987
158
1987
1989 1992
1995
1998
2000
2003
2006 2009
2011
2014
159
160
2014
2017 2019
2022
2025
2028
2030
2033 2036
2038
2041
2041
2044 2047
2049
2052
2055
2057
2060 2063
2066
2068
161
2068
2071 2074
2076
2079
2082
2084
2087 2090
2092
2095
162
2095
2098 2101
2103
2106
2109
2111
2114 2117
2119
2122
163
2122
2125 2127
2130
2133
2135
2138
2140 2143
2146
2148
164
165
2148
2151 2154
2156
2159
2162
2164
2167 2170
2172
2175
2175
2177 2180
2183
2185
2188
2391
2193 2196
2198
2201
166
2201
2204 2206
2209
2212
2214
2217
2219 2222
2225
2227
167
2227
2230 2232
2235
2238
2240
2243
2245 2248
2251
2253
168
2253
2256 2258
2261
2263
2266
2269
2271 2274
2276
2279
169
170
2279
2281 2284
2287
2289
2315
2292
2294
2297 2299
2302
2304
2330
2304
2307 2310
2312
2317
2320
2322 2325
2327
171
2330
2333 2335
2338
2340
2343
2345
2348 2350
2353
2355
172
2355
2358 2360
2363
2305
2368
2370
2373 2375
2378
2380
173
2380
2383 2385
2388
2390
2393
2395
2398 2400
2403
2405
174
175
2405
2408 2410
2413
2415
2418
2420
2423 2425
2428
2430
2430
2433 2435
2438
2440
2443
2445
2448 2450
2453
2455
176
2455
2458 2460
2463
2465
2467
2470
2472 2475
2477
2480
177
2480
2482 2485
2487
2490
2492
2494
2497 2499
2502
2504
178
2504
2507 2509
2512
2514
2516
2519
2521 2524
2526
2529
179
180
2529
2531 2533
2536
2538 2541
2543
2545 2548
2550
2553
2553
2555 2558
2560
2562 '
2565
2567
2570 2572
2574
2577
181
2577
2579 2582
2584
2586
2589
2591
2594 2596
2598
2601
182
2601
2603 2605
2608
2610
2613
2615
2617 2620
2622
2625
183
2625
2627 2629
2632
2634
2636
2639
2641 2643
2646
2648
184
185
2648
2651 2653 2655
2658
2660
2662
2665 2667
2669
2672
2672
2674 2676
2679
2681
2083
2686
2688 2690
2693
2695
186
2695
2697 2700
2702
2704
2707
2709
2711 2714
2716
2718
187
2718
2721 2723
2725
2728
2730
2732
2735 2737
2739
2742
188
2742
2744 2746
2749
2751
2753
2755
2758 2760
2762
2765
189
190
2765
2767 2769
2772
2774
2776
2778
2781 2783
2785
2788
2788
2790 2792
2794
2797
2799
2801
2804 2806
2808
2810
191
2810
2813 2815
2817
2819
2822
2824
2826 2828
2831
2833
192
2833
2835 2838
2840
2842
2844
2847
2849 2851
2853
2856
193
2856
2858 2860
2862
2865
2867
2869
2871 2874
2876
2878
194
195
2878
2880 2882
2885
2887
2889
2891
2894 2896
2898
2900
2900
2903 2905
2907
2909
2911
2914
2916 2918
2920
2923
196
2923
2925 2927
2929
2931
2934
2936
2938 2940
2942
2945
197
2945
2947 2949
2951
2953
2956
2958
2960 2962
2964
2967
198
2967
2969 2971
2973
2975
2978
2980
2982 2984
2986
2989
199
2989
2991 2993
2995
2997
2999
3002
3004 3006
3008
3010
146
TABLES.
Trigonometric Functions.
RADIANS.
DEGREES.
SINES.
COSINES.
TANGENTS.
COTANGENTS.
Nat. Log.
Nat. Log.
Nat. Log.
Nat.
Log.
0.0000
0°00'
.0000 CO
1.0000 0.0000
.0000 00
CO
CO
90° 00'
1.5708
0.0029
10
.0029 7.4637
1.0000 .0000
.0029 7.4637
343.77 2.5363
50
1.5679
0.0058
20
.0058 .7648
1.0000 .0000
.0058 .7648
171.89
.2352
40
1.5650
0.0087
30
.0087 .9408
1.0000 .0000
.0087 .9409
114.59
.0591
30
1.5621
0.0116
40
.0116 8.0658
.9999 .0000
.0116 8.0658
85.940
1.9342
20
1.5592
0.0145
50
.0145 .1627
.9999 .0000
.0145 .1627
68.750
.8373
10
1.5563
0.0175
POO'
.0175 8.2419
.9998 9.9999
.0175 8.2419
57.290
1.7581
89° 00'
1.5533
0.02(H
10
.0204 .3088
.9998 .9999
.0204 .3089
49.104
.6911
50
1.5504
0.0233
20
.0233 .3668
.9997 .9999
.0233 .3669
42.964
.6331
40
1.5475
0.0262
30
.0262 .4179
.9997 .9999
.0262 .4181
38.188
.5819
30
1.5446
0.0291
40
.0291 .4637
.9996 .9998
.0291 .4638
34.368
.5362
20
1.5417
0.0320
50
.0320 .5050
.9995 .9998
.0320 .5053
31.242
.4947
10
1.5388
0.0349
2° 00'
.0349 8.5428
.9994 9.9997
.0349 8.5431
28.636
1.4569
88° 00'
1.5359
0.0378
10
.0378 .5776
.9993 .9997
.0378 .5779
26.432
.4221
50
1.5330
0.0407
20
.0407 .6097
.9992 .9996
.0407 .6101
24.542
.3899
40
1.5301
0.0436
30
.0436 .6397
.9990 .9996
.0437 .6401
22.904
.3599
30
1.5272
0.0465
40
.0465 .6677
.9989 .9995
.0466 .6682
21.470
.3318
20
1.5243
0.0495
50
.0494 .6940
.9988 .9995
.0495 .6945
20.206
.3055
10
1.5213
0.0524
3° 00'
.0523 8.7188
.9986 9.9994
.0524 8.7194
19.081
1.2806
87° 00'
1.5184
0.0553
10
.0552 .7423
.9985 .9993
.0553 .7429
18.075
.2571
50
1.5155
0.0582
20
.0581 .7645
.9983 .9993
.0582 .7652
17.169
.2348
40
1.5126
0.0611
30
.0610 .7857
.9981 .9992
.0612 .7865
16.350
.2135
30
1.5097
0.0640
40
.0640 .8059
.9980 .9991
.0641 .8067
15.605
.1933
20
1.5068
0.0669
50
.0669 .8251
.9978 .9990
.0670 .8261
14.924
.1739
10
1.5039
0.0698
4° 00'
.0698 S.S436
.9976 9.9989
.0699 8.8446
14.301
1.1554
86° 00'
1.5010
0.0727
10
.0727 .8613
.9974 .9989
.0729 .8624
13.727
.1376
50
1.4981
0.0756
20
.0756 .8783
.9971 .9988
.0758 .8795
13.197
.1205
40
1.4952
0.0785
30
.0785 .8946
.9969 .9987
.0787 .8960
12.706
.1040
30
1.4923
0.0814
40
.0814 .9104
.9967 .9986
.0816 .9118
12.251
.0882
20
1.4893
0.0844
50
.0843 .9256
.9964 .9985
.0846 .9272
11.826
.0728
10
1.4864
0.0873
5°00'
.0872 8.9403
.9962 9.9983
.0875 8.9420
11.430
1.0580
85° 00'
1.4835
0.0902
10
.0901 .9545
.9959 .9982
.0904 .9563
11.059
.0«7
50
1.4806
0.0931
20
.0929 .9682
.9957 .9981
.0934 .9701
]0.712
.0299
40
1.4777
0.0960
30
.0958 .9816
.9954 .9980
.0963 .9836
10.385
.0164
30
1.4748
0.0989
40
.0987 .9945
.9951 .9979
.0992 .9966
10.078
.0034
20
1.4719
0.1018
50
.1016 9.0070
.9948 .9977
.1022 9.0093
9.7882 0.9907
10
1.4690
0.1047
6° 00'
.1045 9.0192
.9945 9.9976
.1051 9.0216
9.5144
0.9784
84° 00'
1.4661
0.1076
10
1074 .0311
.9942 .9975
.1080 .0336
9.2553
.9664
50
1.4632
0.1105
20
.1103 .0426
.9939 .9973
.1110 .0453
9.0098
.9547
40
1.4603
0.1134
30
.1132 .0539
.9936 .9972
.1139 .0567
8.7769
.9433
30
1.4574
0.1164
40
.1161 .0648
.9932 .9971
.1169 .0678
8.5555
.9322
20
1.4544
0.1193
50
.1190 .0755
.9929 .9969
.1198 .0786
8.3450
.9214
10
1.4515
0.1222
7° 00'
.1219 9.0859
.9925 9.9968
.1228 9.0891
8.1443
0.9109
83° 00'
1.4486
0.1251
10
.1248 .0961
.9922 .9966
.1257 .0995
7.9530
.9005
50
1.4457
0.1280
20
.1276 .1060
.9918 .9964
.1287 .1096
7.7704
.8904
40
1.4428
0.1309
30
.1305 .1157
.9914 .9963
.1317 .1194
7.5958
.8806
30
1.4399
0.1338
40
.1334 .1252
.9911 .9961
.1346 .1291
7.4287
.8709
20
1.4370
0.1367
50
.1363 .1345
.9907 .9959
.1376 .1385
7.2687
.8615
10
1.4341
0.1396
8° 00'
.1392 9.1436
.9903 9.9958
.1405 9.1478
7.1154
0.8522
82° 00'
1.4312
0.1425
10
.1421 .1525
.9899 .9956
.1435 .1569
6.9682
.8431
50
1.4283
0.1454
20
.1449 .1612
.9894 .9954
.1465 .1658
6.8269
.8342
40
1.4254
0.1484
30
.1478 .1697
.9890 .9952
.1495 .1745
6.6912
.8255
30
1.4224
0.1513
40
.1507 .1781
.9886 .9950
.1524 .1831
6.5606
.8169
20
1.4195
0.1542
50
.1536 .1863
.9881 .9948
.1554 .1915
6.4348
.8085
10
1.4166
0.1571
9° 00'
.1564 9.1943
.9877 9.9946
.1584 9.1997
6.3138 0.8003
81° 00'
1.4137
Nat. Log.
Nat. Log.
Nat. Log.
Nat.
Log.
■^
COSINES.
SINES.
COTANGENTS.
TANGENTS.
DEGREES.
RADIANS.
TABLES.
147
Trigonometric Functions.
RADIANS.
DEGREES.
SINES.
COSINES.
TANGENTS.
COTANGENTS.
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
0.1571
9° 00'
.1564 9.1943
.9877 9.9946
.1584 9.1997
6.3138 0.8003
81° 00'
1.4137
0.1600
10
.1593 .2022
.9872 .9944
.1614 .2078
6.1970 .7922
50
1.4108
0.1629
20
.1622 .2100
.9868 .9942
.1644 .2158
6.0S44 .7842
40
1.4079
0.1658
30
.1650 .2176
.9863 .9940
.1673 .2236
5.9758 .7764
30
1.4050
0.1687
40
.1679 .2251
.9858 .9938
.1703 .2313
5.8708 .7687
20
1.4021
0.1716
50
.1708 .2324
.9853 .9936
.1733 .2389
5.7694 .7611
10
1.3992
0.1745
10° 00'
.1736 9.2397
.9848 9.9934
.1763 9.2463
5.6713 0.7537
80° 00'
1.3963
0.1774
10
.1765 .2468
.9843 .9931
.1793 .2536
5.5764 .7464
50
1.3934
0.1804
20
.1794 .2538
.9838 .9929
.1823 .2609
5.4845 .7391
40
1.3904
0.1833
30
.1822 .2606
.9833 .9927
.1853 .2680
5.3955 .7320
30
1.3875
0.1862
40
.1851 .2674
.9827 .9924
.1883 .2750
5.3093 .7250
20
1.3846
0.1891
50
.1880 .2740
.9822 .9922
.1914 .2819
5.2257 .7181
10
1.3817
0.1920
11° 00'
.1908 9.2806
.9816 9.9919
.1944 9.2S87
5.1446 0.7113
79° 00'
1.3788
0.1949
10
.1937 .2870
.9811 .9917
.1974 .2953
5.0658 .7047
50
1.3759
0.1978
20
.1965 .2934
.9805 .9914
.2004 .3020
4.9894 .6980
40
1.3730
0.2007
30
.1994 .2997
.9799 .9912
.2035 .3085
4.9152 .6915
30
1.3701
0.2036
40
.2022 .3058
.9793 .9909
.2065 .3149
4.8430 .6851
20
1.3672
0.2065
50
.2051 .3119
.9787 .9907
.2095 .3212
4.7729 .6788
10
1.3643
0.2094
12° 00'
.2079 9.3179
.9781 9.9904
.2126 9.3275
4.7046 0.6725
78° 00'
1.3614
0.2123
10
.2108 .3238
.9775 .9901
.2156 .3336
4.6382 .6664
50
1.3584
0.2153
20
.2136 .3296
.9769 .9899
.2186 .3397
4.5736 .6603
40
1.3555
0.2182
30
.2164 .3353
.9763 .9896
.2217 .3458
4.5107 .6542
30
1.3526
0.2211
40
.2193 .3410
.9757 .9893
.2247 .3517
4.4494 .6483
20
1.3497
0.2240
50
.2221 .3466
.9750 .9890
.2278 .3576
4.3897 .6424
10
1.3468
0.2269
13° 00'
.2250 9.3521
.9744 9.9887
.2309 9.3634
4.3315 0.6366
77° 00'
1.3439
0.2298
10
.2278 .3575
.9737 .9884
.2339 .3691
4.2747 .6309
50
1.3410
0.2327
20
.2306 .3629
.9730 .9881
.2370 .3748
4.2193 .6252
40
1.3381
0.2356
30
.2334 .3682
.9724 .9878
.2401 .3804
4.1653 .6196
30
1.3352
0.2385
40
.2363 .3734
.9717 .9875
.2432 .3859
4.1126 .6141
20
1.3323
0.2414
50
.2391 .3786
.9710 .9872
.2462 .3914
4.0611 .6086
10
1.3294
0.2443
14° 00'
.2419 9.3837
.9703 9.9869
.2493 9.3968
4.0108 0.6032
76° 00'
1.3265
0.2473
10
.2447 .3887
.9696 .9866
.2524 .4021
3.9617 .5979
50
1.3235
0.2502
20
.2476 .3937
.9689 .9863
.2555 .4074
3.9136 .5926
40
1.3206
0.2531
30
.2504 .3986
.9681 .9859
.2586 .4127
3.8667 .5873
30
1.3177
0.2560
40
.2532 .4035
.9674 .9856
.2617 .4178
3.8208 .5822
20
1.3148
0.2589
50
.2560 .4083
.9667 .9853
.2648 .4230
3.7760 .5770
10
1.3119
0.2618
15° 00'
.2588 9.4130
.9659 9.9849
.2679 9.4281
3.7321 0.5719
75° 00'
1.3090
0.2647
10
.2616 .4177
.9652 .9846
.2711 .4331
3.6891 .5669
50
1.3061
0.2676
20
.2644 .4223
.9644 .9843
.2742 .4381
3.6470 .5619
40
1.3032
0.2705
30
.2672 .4269
.9636 .9839
.2773 .4430
3.6059 .5570
30
1.3003
0.2734
40
.2700 .4314
.9628 .9836
.2805 .4479
3.5656 .5521
20
1.2974
0.2763
50
.2728 .4359
.9621 .9832
.2836 .4527
3.5261 .5473
10
1.2945
0.2793
16° 00'
.2756 9.4403
.9613 9.9828
.2867 9.4575
3.4874 0.5425
74° 00'
1.2915
0.2822
10
.2784 .4447
.9605 .9825
.2899 .4622
3.4495 .5378
50
1.2886
0.2851
20
.2812 .4491
.9596 .9821
.2931 .4669
3.4124 .5331
40
1.2857
0.2880
30
.2840 .4533
.9588 .9817
.2962 .4716
3.3759 .5284
30
1.2828
0.2909
40
.2868 .4576
.9580 .9814
.2994 .4762
3.3402 .5238
20
1.2799
0.2938
50
.2896 .4618
.9572 .9810
.3026 .4808
3.3052 .5192
10
1.2770
0.2967
17° 00'
.2924 9.4659
.9563 9.9806
.3057 9.4853
3.2709 0.5147
73° 00'
1.2741
0.2996
10
.2952 .4700
.9555 .9802
.3089 .4898
3.2371 .5102
50
1.2712
0.3025
20
.2979 .4741
.9546 .9798
.3121 .4943
3.2041 .5057
40
1.2683
0.3054
30
.3007 .4781
.9537 .9794
.3153 .4987
3.1716 .5013
30
1.2654
0.3083
40
.3035 .4821
.9528 .9790
.3185 .5031
3.1397 .4969
20
1.2625
0.3113
50
.3062 .4861
.9520 .9786
.3217 .5075
3.1084 .4925
10
1.2595
0.3142
18° 00'
.3090 9.4900
.9511 9.9782
.3249 9.5118
3.0777 0.4882
72° 00'
1.2566
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
COSINES.
SINES.
COTANGENTS.
TANGENTS.
DEGREES.
RADIANS.
148
TABLES.
Trigonometric Functions.
RADIANS.
DEGREES.
SINES.
COSINES.
TANGENTS.
COTANGENTS.
Nat. Log.
Nat. Log.
Nat. Log.
i Nat.
Log.
0.3142
18° 00'
.3090 9.4900
.9511 9.9782
.3249 9.5118
3.0777 0.4882
72° 00'
1.2566
0.3171
10
.3118 .4939
.9502 .9778
.3281 .5161
3.0475
.4839
50
1.2537
0.3200
20
.3145 .4977
.9492 .9774
.3314 .5203
3.0178
.4797
40
1.2508
0.3229
30
.3173 .5015
.9483 .9770
.3346 .5245
' 2.9887
.4755
30
1.2479
0.3258
40
.3201 .5052
.9474 .9765
.3378 .5287
2.9600
.4713
20
1.2450
0.3287
50
.3228 .5090
.9465 .9761
.3411 .5329
2.9319
.4671
10
1.2421
0.3316
19° 00'
.3256 9.5126
.9455 9.9757
.3443 9.5370
2.9042
0.4630
71° 00'
1.2392
0.3345
10
.3283 .5163
.9446 .9752
.3476 .5411
2.8770
.4589
50
1.2363
0.3374
20
.3311 .5199
.9436 .9748
.3508 .5451
2.8502
.4549
40
1.2334
0.3403
30
.3338 .5235
.9426 .9743
.3541 .5491
2.8239
.4509
30
1.2305
0.3432
40
.3365 .5270
.9417 .9739
.3574 .5531
2.7980
.4469
20
1.2275
0.3462
50
.3393 .5306
.9407 .9734
.3607 .5571
2.7725
.4429
10
1.2246
0.3491
20° 00'
.3420 9.5341
.9397 9.9730
.3640 9.5611
2.7475
0.4389
70° 00'
1.2217
0.3520
10
.3448 .5375
.9387 .9725
.3673 .5650
2.7228
.4350
SO
1.2188
0.3549
20
.3475 .5409
.9377 .9721
.3706 .5689
2.6985
.4311
40
1.2159
0.3578
30
.3502 .5443
.9367 .9716
.3739 .5727
2.6746
.4273
30
1.2130
0.3607
40
.3529 .5477
.9356 .9711
.3772 .5766
2.6511
.4234
20
1.2101
0.3636
50
.3557 .5510
.9346 .9706
.3805 .5804
2.6279
.4196
10
1.2072
0.3665
21° 00'
.3584 9.5543
.9336 9.9702
.3839 9.5842
2 6051
0.4158
69° 00'
1.2043
0.3694
10
.3611 .5576
.9325 .9697
.3872 .5879
2.5826
.4121
50
1.2014
0.3723
20
.3638 .5609
.9315 .9692
.3906 .5917
2.5605
.4083
40
1.1985
0.3752
30
.3665 .5641
.9304 .9687
.3939 .5954
25386
.4046
30
1.1956
0.3782
40
.3692 .5673
.9293 .9682
.3973 .5991
2.5172
.4009
20
1.1926
0.3811
50
.3719 .5704
.9283 .9677
.4006 .6028
2.4960
.3972
10
1.1897
0.3840
22° 00'
.3746 9.5736
.9272 9.9672
.4040 9.6064
2.4751
0.3936
68° 00'
1.1868
0.3869
10
.3773 .5767
.9261 .9667
.4074 .6100
2.4545
.3900
50
1.1839
0.3898
20
.3800 .5798
.9250 .9661
.4108 .6136
2.4342
.3864
40
1.1810
0.3927
30
.3827 .5828
.9239 .9656
.4142 .6172
2.4142
.3828
30
1.1781
0.3956
40
.3854 .5859
.9228 .9651
.4176 .6208
2.3945
.3792
20
1.1752
0.3985
SO
.3831 .5889
.9216 .9646
.4210 .6243
23750
.3757
10
1.1723
0.4014
23° 00'
.3907 9.S919
.9205 9.9640
.4245 9.6279
2.3559 0.3721
67° 00'
1.1694
0.4043
10
.3934 .5948
.9194 .9635
.4279 .6314
2.3369
.3686
SO
1.1665
0.4072
20
.3961 .5978
.9182 .9629
.4314 .6348
2.3183
.3652
40
1.1636
0.4102
30
.3987 .6007
.9171 .9624
.4348 .6383
2.2998
.3617
30
1.1606
0.4131
40
.4014 , .6036
.9159 .9618
.4383 .6417
2.2817
.3583
20
1.1577
0.4160
50
.4041 .6065
.9147 .9613
.4417 .6452
2.2637
.3548
10
1.1548
0.4189
24° 00'
.4067 9.6093
.9135 9.9607
.4452 9.6486
2.2460 0.3514
66° 00'
1.1519
0.4218
10
.4094 .6121
.9124 .9602
.4487 .6520
2.2286
.3480
SO
1.1490
0.4247
20
.4120 .6149
.9112 .9596
.4522 .6553
2.2113
.3447
40
1.1461
0.4276
30
.4147 .6177
.9100 .9590
.4557 .6587
2.1943
.3413
30
1.1432
0.4305
40
.4173 .6205
.9088 .9584
.4592 .6620
2.1775
.3380
20
1.1403
0.4334
50
.4200 .6232
.9075 .9579
.4628 .6654
2.1609
.3346
10
1.1374
0.4363
25° 00'
.4226 9.6259
.9063 9.9573
.4663 9.6687
2 1445
0.3313
65° 00'
1.1345
0.4392
10
.4253 .6286
.9051 .9567
.4699 .6720
2.1283
.3280
50
1.1316
0.4422
20
.4279 .6313
.9038 .9561
.4734 .6752
2.1123
.3248
40
1.1286
0.4451
30
.4305 .6340
.9026 .9555
.4770 .6785
2.0965
.3215
30
1.1257
0.4480
40
.4331 .6366
.9013 .9549
.4806 .6817
2.0809
.3183
20
1.1228
0.4509
50
.4358 .6392
.9001 .9543
.4841 .6850
2.0655
.3150
10
1.1199
0.4538
26° 00'
.4384 9.6418
.8988 9.9537
.4877 9.6882
2.0503 0.3118
64° 00'
1.1170
0.4567
10
.4410 .6444
.8975 .9530
.4913 .6914
2.0353
.3086
50
1.1141
0.4596
20
.4436 .6470
.8962 .9524
.4950 .6946
2.0204
.3054
40
1.1112
0.4625
30
.4462 .6495
.8949 .9518
.4986 .6977
2.0057
.3023
30
1.1083
0.4654
40
.4488 .6521
.8936 .9512
.5022 .7009
1.9912
.2991
20
1.1054
0.4683
SO
.4514 .6546
.8923 .9505
.5059 .7040
1.9768
.2960
10
1.1025
0.4712
27° 00'
.4540 9.6570
.8910 9.9499
.5095 9.7072
1.9626 0.2928
63° 00'
1.0996
Nat. Log.
Nat. Log.
Nat. Log.
Nat.
Log.
COSINES.
SINES.
COTANGENTS.
TANGENTS.
DEGREES.
RADIANS.
TABLES.
149
Trigonometric Functions.
RADIANS.
DEGREES.
SINES.
COSINES.
TANGENTS.
COTANGENTS.
Nat. Log.
Nat. Log.
Nat.
Log.
Nat. Log.
0.4712
27° 00'
.4540 9.6570
.8910 9.9499
.5095
9.7072
1.9626 0.2928
63° 00'
1.0996
0.4741
10
.4566 .6595
.8897 .9492
.5132
.7103
1.9486 .2897
50
1.0966
0.4771
20
.4592 .6620
.8884 .9486
.5169
.7134
1.9347 .2866
40
1.0937
0.4800
30
.4617 .6644
.8870 .9479
.5206
,7165
1.9210 .2835
30
1.0908
0.4829
40
.4643 .6668
.8857 .9473
.5243
.7196
1.9074 .2804
20
1.0879
0.4858
50
.4669 .6692
.8843 .9466
.5280
.7226
1.8940 .2774
10
1.0850
0.4887
28° 00'
.4695 9.6716
.8829 9.9459
.5317 9.7257
1.8807 0.2743
62° 00'
1.0821
0.4916
10
.4720 .6740
.8816 .9453
.5354
.7287
1.8676 .2713
50
1.0792
0.4945
20
.4746 .6763
.8802 .9446
.5392
.7317
1.8546 .2683
40
1.0763
0.4974
30
.4772 .6787
.8788 .9439
.5430
.7348
1.8418 .2652
30
1.0734
0.5003
40
.4797 .6810
.8774 .9432
.5467
.7378
1.8291 .2622
20
1.0705
0.5032
50
.4823 .6833
.8760 .9425
.5505
.7408
1.S165 .2592
10
1.0676
0.5061
29° 00'
.4848 9.6856
.8746 9.9418
.5543 9.7438
1.8040 0.2562
61° 00'
1.0647
0.5091
10
.4874 .6878
.8732 .9411
.5581
.7467
1.7917 .2533
50
1.0617
0.5120
20
.4899 .6901
.8718 .9404
.5619
.7497
1.7796 .2503
40
1.0588
0.5149
30
.4924 .6923
.8704 .9397
.5658
.7526
1.7675 .2474
30
1.0559
0.5178
40
.4950 .6946
.8689 .9390
.5696
.7556
1.7556 .2444
20
1.0530
0.5207
50
.4975 .6968
.8675 .9383
.5735
.7585
1.7437 .2415
10
1.0501
0.5236
30° 00'
.5000 9.6990
.8660 9.9375
.5774 9.7614
1.7321 0.2386
60° 00'
1.0472
0.5265
10
.5025 .7012
.8646 .9368
.5812
.7644
1.7205 .2356
50
1.0443
0.5294
20
.5050 .7033
.8631 .9361
.5851
.7673
1.7090 .2327
40
1.0414
0.5323
30
.5075 .7055
.8616 .9353
.5890
.7701
1.6977 .2299
30
1.0385
0.5352
40
.5100 .7076
.8601 .9346
.5930
.7730
1.6864 .2270
20
1.0356
0.5381
50
.5125 .7097
.8587 .9338
.5969
.7759
1.6753 .2241
10
1.0327
0.5411
31° 00'
.5150 9.7118
.8572 9.9331
.6009 9.7788
1.6643 0.2212
59° 00'
1.0297
0.5440
10
.5175 .7139
.8557 .9323
.6048
.7816
1.6534 .2184
50
1.0268
0.5469
20
.5200 .7160
.8542 .9315
.6088
.7845
1.6426 .2155
40
1.0239
0.5498
30
.5225 .7181
.8526 .9.308
.6128
.7873
1.6319 .2127
30
1.0210
0.5527
40
.5250 .7201
.8511 .9300
.6168
.7902
1.6212 .2098
20
1.0181
0.5556
50
.5275 .7222
.8496 .9292
.6208
.7930
1.6107 .2070
10
1.0152
0.5585
32° 00'
.5299 9.7242
.8480 9.9284
.6249 9.7958
1.6003 0.2042
58° 00'
1.0123
0.5614
10
.5324 .7262
.8465 .9276
.6289
.7986
1.5900 .2014
50
1.0094
0.5643
20
.5348 .7282
.8450 .9268
.6330
.8014
1.5798 .1986
40
1.0065
0.5672
30
.5373 .7302
.8434 .9260
.6371
.8042
1.5697 .1958
30
1.0036
0.5701
40
.5398 .7322
.8418 .9252
.6412
.8070
1.5597 .1930
20
1.0007
0.5730
50
.5422 .7342
.8403 .9244
.6453
.8097
1.5497 .1903
10
0.9977
0.5760
33° 00'
.5446 9.7361
.8387 9.9236
.6494 9.8125
1.5399 0.1875
57° 00'
0.9948
0.5789
10
.5471 .7380
.8371 .9228
.6536
.8153
1.5301 .1847
50
0.9919
0.5818
20
.5495 .7400
.8355 .9219
.6577
.8180
1.5204 .1820
40
0.9890
0.5847
30
.5519 .7419
.8339 .9211
.6619
.8208
1.5108 .1792
30
0.9861
0.5876
40
.5544 .7438
.8323 .9203
.6661
.8235
1.5013 .1765
20
0.9832
0.5905
50
.5568 .7457
.8307 .9194
.6703
.8263
1.4919 .1737
10
0.9803
0.5934
34° 00'
.5592 9.7476
.8290 9.9186
.6745
9.8290
1.4826 0.1710
56° 00'
0.9774
0.5963
10
.5616 .7494
.8274 .9177
.6787
.8317
1.4733 .1683
50
0.9745
0.5992
20
.5640 .7513
.8258 .9169
.6830
.8344
1.4641 .1656
40
0.9716
0.6021
30
.5664 .7531
.8241 .9160
.6873
.8371
1.4550 .1629
30
0.9687
0.6050
40
.5688 .7550
.8225 .9151
.6916
.8398
1.4460 .1602
20
0.9657
0.6080
50
.5712 .7568
.8208 .9142
.6959
.8425
1.4370 .1575
10
0.9628
0.6109
35° 00'
.5736 9.7586
.8192 9.9134
.7002
9.8452
1.4281 0.1548
55° 00'
0.9599
0.6138
10
.5760 .7604
.8175 .9125
.7046
.8479
1.4193 .1521
50
0.9570
0.6167
20
.5783 .7622
.8158 .9116
.7089
.8506
1.4106 .1494
40
0.9541
0.6196
30
.5807 .7640
.8141 .9107
.7133
.8533
1.4019 .1467
30
0.9512
0.6225
40
.5831 .7657
.8124 .9098
.7177
.8559
1.3934 .1441
20
0.9483
0.6254
50
.5854 7675
.8107 .9089
.7221
.8586
1.3848 .1414
10
0.9454
0.6283
36° 00'
.5878 9.7692
.8090 9.9080
.7265
9.8613
1.3764 0.1387
54° 00'
0.9425
Nat. Log.
Nat. Log.
Nat.
Log.
Nat. Log.
COSINES.
SINES.
COTANGENTS.
TANGENTS.
DEGREES.
RADIANS.
150
TABLES.
Trigonometric Functions.
RADIANS.
DEGREES.
SINES.
COSINES.
TANGENTS.
COTANGENTS.
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
0.62S3
36° 00'
.5878 9.7692
.8090 9.9080
.7265 9.8613
1.3764 0.13S7
54° 00'
0.9425
0.6312
10
.5901 .7710
.8073 .9070
.7310 .8639
1.3680 .1361
50
0.9396
0.6341
20
.5925 .7727
.8056 .9061
.7355 .8666
1.3597 .1334
40
0.9367
0.6370
30
.5948 .7744
.8039 .9052
.7400 .8692
1.3514 .1308
30
0.9338
0.6400
40
.5972 .7761
.8021 .9042
.7445 .8718
1.3432 .1282
20
0.9308
0.6429
50
.5995 .7778
.8004 .9033
.7490 .8745
1.3351 .1255
10
0.9279
0.6458
37° 00'
.6018 9.7795
.7986 9.9023
.7536 9.8771
1.3270 0.1229
53° 00'
0.9250
0.64S7
10
.6041 .7811
.7969 .9014
.7581 .8797
1.3190 .1203
50
0.9221
0.6516
20
.6065 .7828
.7951 .9004
.7627 .8824
1.3111 .1176
40
0.9192
0.6545
30
.6088 .7844
.7934 .8995
.7673 .8850
1.3032 .1150
30
0.9163
0.6574
40
.6111 .7861
.7916 .8985
.7720 .8876
1.2954 .1124
20
0.9134
0.6603
50
.6134 .7877
.7898 .8975
.7766 .8902
1.2876 .1098
10
0.9105
0.6632
38° 00'
.6157 9.7893
.7880 9.8965
.7813 9.8928
1.2799 0.1072
52° 00'
0.9076
0.6661
10
.6180 .7910
.7862 .8955
.7860 .8954
1.2723 .1046
50
0.9047
0.6690
20
.6202 .7926
.7844 .8945
.7907 .8980
1.2647 .1020
40
0.9018
0.6720
30
.6225 .7941
.7826 .8935
.7954 .9006
1.2572 .0994
30
0.8988
0.6749
40
.6248 .7957
.7808 .8925
.8002 .9032
1.2497 .0968
20
0.8959
0.67 7S
50
.6271 .7973
.7790 .8915
.8050 .9058
1.2423 .0942
10
0.8930
0.6807
39° 00'
.6293 9.7989
.7771 9.8905
.8098 9.9084
1.2349 0.0916
51° 00'
0.8901
0.6836
10
.6316 .8004
.7753 .8895
.8146 .9110
1.2276 .0890
50
0.8872
0.6865
20
.6338 .8020
.7735 .8884
.8195 .9135
1.2203 .0865
40
0.8843
0.6894
30
.6361 .8035
.7716 .8874
.8243 .9161
1.2131 .0839
30
0.8814
0.6923
40
.6383 .8050
.7698 .8864
.8292 .9187
1.2059 .0813
20
0.8785
0.6952
50
.6406 .8066
.7679 .8853
.8342 .9212
1.1988 .0788
10
0.8756
0.6981
40° 00'
.6428 9.S081
.7660 9.8843
.8391 9.9238
1.1918 0.0762
50° 00'
0.8727
0.7010
10
.6450 .8096
.7642 .8832
.8441 .9264
1.1847 .0736
50
0.S698
0.7039
20
.6472 .8111
.7623 .8821
.8491 .9289
1.1778 .0711
40
0.8668
0.7069
30
.6494 .8125
.7604 .8810
.8541 .9315
1.1708 .0685
30
0.8639
0.7098
40
.65.'.7 .8140
.7585 .8800
.8591 .9341
1.1640 .0659
20
0.S610
0.7127
50
.6539 .8155
.7566 .8789
.8642 .9366
1.1571 .0634
10
0.8581
0.7156
41° 00'
.6561 9.8169
.7547 9.8778
.8693 9.9392
1.1504 0.0608
49° 00'
0.8552
0.7185
10
.6583 .8184
.7528 .8767
.8744 .9417
1.1436 .0583
50
0.8523
0.7214
20
.6604 .8198
.7509 .8756
.8796 .9443
1.1369 .0557
40
0.8494
0.7243
30
.6626 .8213
.7490 .8745
.8847 .9468
1.1303 .0532
30
0.8465
0.7272
40
.6648 .8227
.7470 .8733
.8899 .9494
1.1237 .0506
20
0.8436
0.7301
50
.6670 .8241
.7451 .8722
.8952 .9519
1.1171 .0481
10
0.8407
0.7330
42° 00'
.6691 9.8255
.7431 9.8711
.9004 9.9544
1.1106 0.0456
48° 00'
0.8378
0.7359
10
.6713 .8269
.7412 .8699
.9057 .9570
1.1041 .0430
50
0.8348
0.7389
20
.6734 .8283
.7392 .8688
.9110 .9595
1.0977 .0405
40
0.8319
0.7418
30
.6756 .8297
.7373 .8676
.9163 .9621
1.0913 .0379
30
0.8290
0.7447
40
.6777 .8311
.7353 .8665
.9217 .9646
1.0850 .0354
20
0.8261
0.7476
50
.6799 .8324
.7333 .8653
.9271 .9671
1.0786 .0329
10
0.8232
0.7505
43° 00'
.6820 9.8338
.7314 9.8641
.9325 9.9697
1.0724 0.0303
47° 00'
0.8203
0.7534
10
.6841 .8351
.7294 .8629
.9380 .9722
1.0661 .0278
50
0.8174
0.7563
20
.6862 .8365
.7274 .8618
.9435 .9747
1.0599 .0253
40
0.8145
0.7592
30
.6884 .8378
.7254 .8606
.9490 .9772
1.0538 .0228
30
0.8116
0.7621
40
.6905 .8391
.7234 .8594
.9545 .9798
1.0477 .0202
20
0.8087
0.7650
50
.6926 .8405
.7214 .8582
.9601 .9823
l.ail6 .0177
10
0.8058
0.7679
44° 00'
.6947 9.8418
.7193 9.8569
.9657 9.9848
1.0355 0.0152
46° 00'
0.8029
0.7709
10
.6967 .8431
.7173 .8557
.9713 .9874
1.0295 .0126
50
0.7999
0.7738
20
.6988 .8444
.7153 .8545
.9770 .9899
1.0235 .0101
40
0.7970
0.7767
30
.7009 .8457
.7133 .8532
.9827 .9924
1.0176 .0076
30
0.7941
0.7796
40
.7030 .8469
.7112 .8520
.9884 .9949
1.0117 .0051
20
0.7912
0.7825
50
.7050 .8482
.7092 .8507
.9942 .9975
1.0058 .0025
10
0.7883
0.7854
45° 00'
.7071 9.8495
.7071 9.8495
1.0000 0.0000
1.0000 0.0000
45° 00'
0.7854
Nat. Log.
Nat. Log.
Nat. Log.
Nat. Log.
COSINES.
SINES.
COTANGENT.S.
TANGENTS.
DEGREES.
RADIANS.
TABLES.
151
Equivalents of Radians in Degrees, Minutes, and Seconds of Arc.
RAmATJS-
EQUIVALENTS.
RADIANS.
EQUIVALENTS.
0.0001
0° 0' 20".6 or 0°.005730
0.0600
3° 26' 15".9
or 3°.437747
0.0002
0° 0'41".3 or 0°.011459
0.0700
4° 0'3S".5
or 4°.010705
0.0003
0° 1'01".9 or 0°.017189
O.OSOO
4°35'01".2
or 4°.5S3662
0.0004
0° V 22".5 or 0°. 022918
0.0900
5° 9'23".S
or 5°.156620
0.0005
0° 1' 43". 1 or 0°.028648
0.1000
5° 43' 46". 5
or 5°.729578
0.0006
0° 2'03".8 or 0°.034377
0.2000
11°27'33".0
or 11°.459156
0.0007
Qo 2'24".4 or 0°.040107
0.3000
17°11'19".4
or 17°. 188734
0.0008
0° 2'45".0 or 0°.045837
0.4000
22°55'05".9
or 22°.918312
0.0009
0° 3'05".6 or 0°.051566
0.5000
2S°38'52".4
or 28°.647890
0.0010
0° 3'26".3 or 0°. 057296
0.6000
34° 22' 3S".9
or 34°377468
0.0020
0° 6'52".5 or 0°.114S92
0.7000
40° 6'25".4
or 40°. 107046
0.0030
0°10'1S".8 or 0°.171887
O.SOOO
45°50'11".8
or 45°.836624
0.0040
0°13'45".l or 0°. 229183
0.9000
51° 33' 58".3
or 51°.566202
0.0050
0°17'11".3 or 0°.286479
1.0000
57°17'44".8
or 57°.295780
0.0060
0°20'37".6 or 0^.343775
2.0000
114° 35' 29".6
or 1M°.591559
0.0070
0°24'03".9 or 0°.401070
3.0000
171° 53' 14".4
or 171°.8S7339
0.00S0
0°27'30".l or 0°. 458366
4.0000
229° 10' 59".2
or 229°. 183 118
0.0090
0°30'S6".4 or 0°.515662
5.0000
286°28'44".0
or 286°.478898
0.0100
0°34'22".6 or 0°.572958
6.0000
343°46'28".8
or 343°.774677
0.0200
1° 8'45".3 or P.145916
7.0000
401° 4' 13" 6
or 401°.070457
0.0300
1''43'07".9 or 10.718873
8.0000
458° 21' 58".4
or 458°.366236
0.0400
2°17'30".6 or 2°.291831
9.0000
515°39'43".3
or 515°.662016
0.0500
2° 51' S3".2 or 2°.864789
10.0000
572° 57' 2S".l
or 572°.95779S
The Values in Circular Measure of Angles which are given in
Degrees and Minutes.
r
0.0003
9'
0.0026
3°
0.0524
20°
0.3491
100°
\.7453
2'
0.0006
10'
0.0029
4°
0.0698
30°
0.5236
110°
1.9199
3'
0.0009
20'
0.0058
5°
0.0873
40°
0.6981
120°
2.0944
4'
0.0012
30'
0.0087
6°
0.1047
50°
0.8727
130°
2.2689
5'
0.0015
40'
0.0116
7°
0.1222
60°
1.0472
140°
2.4435
6'
0.0017
50'
0.0145
8°
0.1396
70°
1.2217
150°
2.6180
7'
0.0020
1°
0.0175
9°
0.1571
80°
1.3963
160°
2.7925
8'
0.0023
2°
0.0349
10°
0.1745
90°
1.5708
170°
2.9671
PAGE INDEX.
INTEGRALS.
Fundamental forms
Rational algebraic expressions involving (a + bx) and (a' + b'x)
{a + bx") .
" " " (a + bx + cx^) .
(a' + 6'x)and(a + 6a; + cx2)
Rational fractions .........
Irrational algebraic expressions involving Va + bx or \/a + bx .
(1
11
Miscellaneous algebraic expressions
General transcendental forms ......
Expressions involving simple direct trigonometric functions
Expressions involving inverse trigonometric functions
Exponential forms .....
Logarithmic forms .....
Expressions involving hyperbolic functions
Miscellaneous definite integrals
Elliptic integrals
Pages
3,4
5,7
8,9
10,11
11-13
13,14
16,17
18,19
20-23
31
23-27
(a' + b'x) and Va + bx + cx^ 27-30
32-34
35-37
38-51
51-53
63-56
66-58
68-61
62-65
66-72
V a + bx and Va' + b' x
•V x2 ± a2 or Va2 — x^
V2
ax — x^
Va + bx + cx2
AUXILIARY FORMULAS AND TABLES.
Trigonometric functions .
Hyperbolic functions
Elliptic functions, Bessel's functions
Series
Derivatives ....
Green's Theorem and allied formulas
Table of mathematical constants
General formulas of integration
Note on interpolation
Table of the probability integral
Tables of elliptic integrals
Table of hyperbolic functions .
Table of values of e-^
Table of common logarithms of e^ and e-^
Five-place table of natural logarithms
Table of logarithms of T {x) .
Three-place table of natural trigonometric functions
Four-place table of common logarithms of numbers
Four-place table of trigonometric functions
Tables for reducing radians to degrees
152
73-80
81-83
84-87
88-96
97-106
106-109
109
110-114
115
116-120
121-123
124-127
127
128, 129
130-139
140
141
142-145
146-150
151
14 DAY USE
RETURN TO DESK FROM WHICH BORROWED
LOAN DEPT.
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on the date to which renewed.
Renewed books are subject to immediate recall.
NOV 8-1966 6 3
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..Mr^ ' Bb-^
^
\J'u^'
L.<;^'/^iH
ij^fur
M^^m^
AUG 21 1984
CIRCULAT'^M nFPT.
LD 21A-60ni-7,'66
(G4427sl0)476B
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